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Null Hypothesis: Definition, Rejecting & Examples

By Jim Frost 6 Comments

What is a Null Hypothesis?

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

• Null Hypothesis H 0 : No effect exists in the population.
• Alternative Hypothesis H A : The effect exists in the population.

In every study or experiment, researchers assess an effect or relationship. This effect can be the effectiveness of a new drug, building material, or other intervention that has benefits. There is a benefit or connection that the researchers hope to identify. Unfortunately, no effect may exist. In statistics, we call this lack of an effect the null hypothesis. Researchers assume that this notion of no effect is correct until they have enough evidence to suggest otherwise, similar to how a trial presumes innocence.

In this context, the analysts don’t necessarily believe the null hypothesis is correct. In fact, they typically want to reject it because that leads to more exciting finds about an effect or relationship. The new vaccine works!

You can think of it as the default theory that requires sufficiently strong evidence to reject. Like a prosecutor, researchers must collect sufficient evidence to overturn the presumption of no effect. Investigators must work hard to set up a study and a data collection system to obtain evidence that can reject the null hypothesis.

Related post : What is an Effect in Statistics?

Null Hypothesis Examples

Null hypotheses start as research questions that the investigator rephrases as a statement indicating there is no effect or relationship.

After reading these examples, you might think they’re a bit boring and pointless. However, the key is to remember that the null hypothesis defines the condition that the researchers need to discredit before suggesting an effect exists.

Let’s see how you reject the null hypothesis and get to those more exciting findings!

When to Reject the Null Hypothesis

So, you want to reject the null hypothesis, but how and when can you do that? To start, you’ll need to perform a statistical test on your data. The following is an overview of performing a study that uses a hypothesis test.

The first step is to devise a research question and the appropriate null hypothesis. After that, the investigators need to formulate an experimental design and data collection procedures that will allow them to gather data that can answer the research question. Then they collect the data. For more information about designing a scientific study that uses statistics, read my post 5 Steps for Conducting Studies with Statistics .

After data collection is complete, statistics and hypothesis testing enter the picture. Hypothesis testing takes your sample data and evaluates how consistent they are with the null hypothesis. The p-value is a crucial part of the statistical results because it quantifies how strongly the sample data contradict the null hypothesis.

When the sample data provide sufficient evidence, you can reject the null hypothesis. In a hypothesis test, this process involves comparing the p-value to your significance level .

Rejecting the Null Hypothesis

Reject the null hypothesis when the p-value is less than or equal to your significance level. Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. For a mnemonic device, remember—when the p-value is low, the null must go!

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

Failing to Reject the Null Hypothesis

Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. The sample data provides insufficient data to conclude that the effect exists in the population. When the p-value is high, the null must fly!

Note that failing to reject the null is not the same as proving it. For more information about the difference, read my post about Failing to Reject the Null .

That’s a very general look at the process. But I hope you can see how the path to more exciting findings depends on being able to rule out the less exciting null hypothesis that states there’s nothing to see here!

Let’s move on to learning how to write the null hypothesis for different types of effects, relationships, and tests.

Related posts : How Hypothesis Tests Work and Interpreting P-values

How to Write a Null Hypothesis

The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter . Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests.

Related posts : Descriptive vs. Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

Group Means

T-tests and ANOVA assess the differences between group means. For these tests, the null hypothesis states that there is no difference between group means in the population. In other words, the experimental conditions that define the groups do not affect the mean outcome. Mu (µ) is the population parameter for the mean, and you’ll need to include it in the statement for this type of study.

For example, an experiment compares the mean bone density changes for a new osteoporosis medication. The control group does not receive the medicine, while the treatment group does. The null states that the mean bone density changes for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group means are equal in the population: µ 1 = µ 2 , or µ 1 – µ 2 = 0
• Alternative Hypothesis H A : Group means are not equal in the population: µ 1 ≠ µ 2 , or µ 1 – µ 2 ≠ 0.

Group Proportions

Proportions tests assess the differences between group proportions. For these tests, the null hypothesis states that there is no difference between group proportions. Again, the experimental conditions did not affect the proportion of events in the groups. P is the population proportion parameter that you’ll need to include.

For example, a vaccine experiment compares the infection rate in the treatment group to the control group. The treatment group receives the vaccine, while the control group does not. The null states that the infection rates for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group proportions are equal in the population: p 1 = p 2 .
• Alternative Hypothesis H A : Group proportions are not equal in the population: p 1 ≠ p 2 .

Correlation and Regression Coefficients

Some studies assess the relationship between two continuous variables rather than differences between groups.

In these studies, analysts often use either correlation or regression analysis . For these tests, the null states that there is no relationship between the variables. Specifically, it says that the correlation or regression coefficient is zero. As one variable increases, there is no tendency for the other variable to increase or decrease. Rho (ρ) is the population correlation parameter and beta (β) is the regression coefficient parameter.

For example, a study assesses the relationship between screen time and test performance. The null states that there is no correlation between this pair of variables. As screen time increases, test performance does not tend to increase or decrease.

• Null Hypothesis H 0 : The correlation in the population is zero: ρ = 0.
• Alternative Hypothesis H A : The correlation in the population is not zero: ρ ≠ 0.

For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.

The preceding examples are all for two-tailed hypothesis tests. To learn about one-tailed tests and how to write a null hypothesis for them, read my post One-Tailed vs. Two-Tailed Tests .

Related post : Understanding Correlation

Neyman, J; Pearson, E. S. (January 1, 1933).  On the Problem of the most Efficient Tests of Statistical Hypotheses .  Philosophical Transactions of the Royal Society A .  231  (694–706): 289–337.

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January 11, 2024 at 2:57 pm

Thanks for the reply.

January 10, 2024 at 1:23 pm

Hi Jim, In your comment you state that equivalence test null and alternate hypotheses are reversed. For hypothesis tests of data fits to a probability distribution, the null hypothesis is that the probability distribution fits the data. Is this correct?

January 10, 2024 at 2:15 pm

Those two separate things, equivalence testing and normality tests. But, yes, you’re correct for both.

Hypotheses are switched for equivalence testing. You need to “work” (i.e., collect a large sample of good quality data) to be able to reject the null that the groups are different to be able to conclude they’re the same.

With typical hypothesis tests, if you have low quality data and a low sample size, you’ll fail to reject the null that they’re the same, concluding they’re equivalent. But that’s more a statement about the low quality and small sample size than anything to do with the groups being equal.

So, equivalence testing make you work to obtain a finding that the groups are the same (at least within some amount you define as a trivial difference).

For normality testing, and other distribution tests, the null states that the data follow the distribution (normal or whatever). If you reject the null, you have sufficient evidence to conclude that your sample data don’t follow the probability distribution. That’s a rare case where you hope to fail to reject the null. And it suffers from the problem I describe above where you might fail to reject the null simply because you have a small sample size. In that case, you’d conclude the data follow the probability distribution but it’s more that you don’t have enough data for the test to register the deviation. In this scenario, if you had a larger sample size, you’d reject the null and conclude it doesn’t follow that distribution.

I don’t know of any equivalence testing type approach for distribution fit tests where you’d need to work to show the data follow a distribution, although I haven’t looked for one either!

February 20, 2022 at 9:26 pm

Is a null hypothesis regularly (always) stated in the negative? “there is no” or “does not”

February 23, 2022 at 9:21 pm

Typically, the null hypothesis includes an equal sign. The null hypothesis states that the population parameter equals a particular value. That value is usually one that represents no effect. In the case of a one-sided hypothesis test, the null still contains an equal sign but it’s “greater than or equal to” or “less than or equal to.” If you wanted to translate the null hypothesis from its native mathematical expression, you could use the expression “there is no effect.” But the mathematical form more specifically states what it’s testing.

It’s the alternative hypothesis that typically contains does not equal.

There are some exceptions. For example, in an equivalence test where the researchers want to show that two things are equal, the null hypothesis states that they’re not equal.

In short, the null hypothesis states the condition that the researchers hope to reject. They need to work hard to set up an experiment and data collection that’ll gather enough evidence to be able to reject the null condition.

February 15, 2022 at 9:32 am

Dear sir I always read your notes on Research methods.. Kindly tell is there any available Book on all these..wonderfull Urgent

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Null Hypothesis Definition and Examples, How to State

What is the null hypothesis, how to state the null hypothesis, null hypothesis overview.

Why is it Called the “Null”?

The word “null” in this context means that it’s a commonly accepted fact that researchers work to nullify . It doesn’t mean that the statement is null (i.e. amounts to nothing) itself! (Perhaps the term should be called the “nullifiable hypothesis” as that might cause less confusion).

Why Do I need to Test it? Why not just prove an alternate one?

The short answer is, as a scientist, you are required to ; It’s part of the scientific process. Science uses a battery of processes to prove or disprove theories, making sure than any new hypothesis has no flaws. Including both a null and an alternate hypothesis is one safeguard to ensure your research isn’t flawed. Not including the null hypothesis in your research is considered very bad practice by the scientific community. If you set out to prove an alternate hypothesis without considering it, you are likely setting yourself up for failure. At a minimum, your experiment will likely not be taken seriously.

• Null hypothesis : H 0 : The world is flat.
• Alternate hypothesis: The world is round.

Several scientists, including Copernicus , set out to disprove the null hypothesis. This eventually led to the rejection of the null and the acceptance of the alternate. Most people accepted it — the ones that didn’t created the Flat Earth Society !. What would have happened if Copernicus had not disproved the it and merely proved the alternate? No one would have listened to him. In order to change people’s thinking, he first had to prove that their thinking was wrong .

How to State the Null Hypothesis from a Word Problem

You’ll be asked to convert a word problem into a hypothesis statement in statistics that will include a null hypothesis and an alternate hypothesis . Breaking your problem into a few small steps makes these problems much easier to handle.

Step 2: Convert the hypothesis to math . Remember that the average is sometimes written as μ.

H 1 : μ > 8.2

Broken down into (somewhat) English, that’s H 1 (The hypothesis): μ (the average) > (is greater than) 8.2

Step 3: State what will happen if the hypothesis doesn’t come true. If the recovery time isn’t greater than 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or less than 8.2 weeks.

H 0 : μ ≤ 8.2

Broken down again into English, that’s H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2

How to State the Null Hypothesis: Part Two

But what if the researcher doesn’t have any idea what will happen.

Example Problem: A researcher is studying the effects of radical exercise program on knee surgery patients. There is a good chance the therapy will improve recovery time, but there’s also the possibility it will make it worse. Average recovery times for knee surgery patients is 8.2 weeks. 

Step 1: State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.

H 0 : μ = 8.2

Broken down into English, that’s H 0 (The null hypothesis): μ (the average) = (is equal to) 8.2

Step 2: Figure out the alternate hypothesis . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?

H 1 : μ ≠ 8.2

In English again, that’s H 1 (The  alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2

That’s How to State the Null Hypothesis!

Check out our Youtube channel for more stats tips!

Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley.

Econometrics for Business Analytics

Chapter 7 hypothesis testing.

Hypothesis testing is the most important thing you learned in business statistics. It is the foundation of the statistical world.

Hypothesis testing tells us if the treatment effect we observed is statistically significant .

A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses.

7.1 Statistical Hypotheses

The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected.

There are two types of statistical hypotheses.

• Null hypothesis. The null hypothesis, denoted by Ho, is usually the hypothesis that sample observations result purely from chance.
• Alternative hypothesis. The alternative hypothesis, denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

7.2 Case Study: Birthweight and Smoking

There is a lot of evidence that smoking is bad for one’s health. What is less certain is the effect of smoking on birthweight.

You might ask, “how is this hard to measure or why is it controversial?”

The issue is with reporting. If you are a pregnant mother, how honestly would you respond to the question of “Do you smoke?”

It is easy to see that mothers may lie about how much or even if they smoked while pregnant.

7.2.1 Load the Data

First, let’s load the data.

7.2.2 Difference in Birthweight by Smoking Status

Compare birthweight by smoking status, we can see that smoker babies are smaller, but there is overlap.

7.2.3 Differences in Birthweight by Smoking Status

7.2.4 Differences in Birthweight by Smoking Status

How can we assess whether this difference is statistically significant?

Let’s compute a summary table

7.2.5 Differences in Birthweight by Smoking Status

The standard deviation is good to have, but to assess statistical significance we really want to have the standard error.

If we use a confidence interval around the sample means, there is less overlap between the two groups. \[\bar{x}\pm se*t_{\alpha /2} \]

7.2.6 T-test for Birthweight by Smoking Status

In this case study, we have been looking at a sample of mothers, some who smoke and some who do not. These are samples and not populations. Therefore, we need to use a two sample t-test.

This difference is looking quite significant. To run a two-sample t-test, we can simple use the t.test() function.

7.2.7 Interpreting Output

There are a few things from the output we can note.

First, is the p-value. The p-value tells us the likelihood that the null hypothesis (in this case no difference between groups) is true. For p-values less than 5 percent, we can reject the null hypothesis and state there is a statistically significant difference between the two groups.

The p-value in our t-test was 0.0070025, which is less than 1 percent so we can reject the null hypothesis.

Our study finds that birth weights are on average higher in the non-smoking group compared to the smoking group (t-statistic 2.73, p=0.007, 95 % CI [78.6, 489]g)

7.3 Standard Levels of significance

Levels of significance, \(\alpha\) , are commonly - \(\alpha\) = 0.10 is marginally significant - \(\alpha\) = 0.05 is significant - \(\alpha\) = 0.01 is very significant

We reject the null hypothesis \(H_0\) if the p-value \(< \alpha\) .

The significance level represents the probability of committing a Type I error that we are willing to accept. A Type I error is rejecting the null hypothesis when the null hypothesis is true.

7.4 Warning

7.4.1 can we accept the null hypothesis.

Some researchers say that a hypothesis test can have one of two outcomes: you accept the null hypothesis or you reject the null hypothesis. Many statisticians, however, take issue with the notion of “accepting the null hypothesis.” Instead, they say: you reject the null hypothesis or you fail to reject the null hypothesis.

Why the distinction between “acceptance” and “failure to reject?” Acceptance implies that the null hypothesis is true. Failure to reject implies that the data are not sufficiently persuasive for us to prefer the alternative hypothesis over the null hypothesis.

Think of it this way. In court, we say a person is either guilty or not guilty. We do not say the person is innocent. That is, we conclude that either there is enough evidence to say the person is guilty or there isn’t enough evidence (fail to reject).

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In this course, you will learn why it is rational to use the parameters recovered under the Classical Linear Regression Model for hypothesis testing in uncertain contexts. You will:

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• Efficiency • 1 minute
• Consistency • 1 minute

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• Understanding Efficiency • 10 minutes
• Understanding Consistency • 10 minutes

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• Check Your Understanding of Efficiency • 30 minutes
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3 discussion prompts • Total 30 minutes

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4 videos 6 readings 7 quizzes 1 discussion prompt 2 ungraded labs

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• The t-Test • 4 minutes
• The F-Test • 5 minutes
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6 readings • Total 60 minutes

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• Exploring the Test of Significance • 10 minutes
• Example of the t-Test • 10 minutes
• Test Joint Hypothesis • 10 minutes
• An Example • 10 minutes
• Types of Errors • 10 minutes

7 quizzes • Total 180 minutes

• Knowledge Check: Hypothesis Testing • 20 minutes
• Building a Hypothesis • 10 minutes
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• Differences between t and F-Tests • 30 minutes
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1 discussion prompt • Total 10 minutes

• The Importance of Hypothesis Testing • 10 minutes

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4 discussion prompts • Total 40 minutes

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2 videos 7 readings 6 quizzes 1 peer review 4 discussion prompts 3 ungraded labs

2 videos • Total 5 minutes

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7 readings • Total 70 minutes

• Consequences of the Violations • 10 minutes
• Congratulations • 10 minutes

6 quizzes • Total 170 minutes

• Understanding Hypothesis Testing • 30 minutes
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• Understanding Heteroscedasticity • 30 minutes
• Understanding Autocorrelation • 30 minutes
• Understanding Stochastic Regressors • 30 minutes
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1 peer review • Total 120 minutes

• Hypothesis and Diagnostic Testing • 120 minutes
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Econometric Theory/Statistical Inference/Hypothesis Testing

Basic concepts.

To conduct a successful hypothesis test, the following are required:

• Testable Hypothesis

• Feasible test statistic

A test statistic is a random variable whose value for given sample data determines whether the null is rejected or retained. It is feasible when:

• Its value can be calculated from the given sample data
• Decision rule

A decision rule clearly delineates the:

Procedure for testing a hypothesis

• Specify the test statistic and its distribution.
• Select a significance level (α) and determine the corresponding critical values (for the particular distribution).
• Apply the decision rule and state the conclusion (or inference) implied by the sample value of the test statistic.
• The null hypothesis will always be the position where there is an equality (either strong or weak), and the alternate hypothesis will have the inequality.

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Econometrics: Definition, Models, and Methods

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

Econometrics is the use of statistical and mathematical models to develop theories or test existing hypotheses in economics and to forecast future trends from historical data. It subjects real-world data to statistical trials and then compares the results against the theory being tested.

Depending on whether you are interested in testing an existing theory or in using existing data to develop a new hypothesis, econometrics can be subdivided into two major categories: theoretical and applied. Those who routinely engage in this practice are commonly known as econometricians.

Key Takeaways

• Econometrics is the use of statistical methods to develop theories or test existing hypotheses in economics or finance.
• Econometrics relies on techniques such as regression models and null hypothesis testing.
• Econometrics can also be used to try to forecast future economic or financial trends.
• As with other statistical tools, econometricians should be careful not to infer a causal relationship from statistical correlation.
• Some economists have criticized the field of econometrics for prioritizing statistical models over economic reasoning.

Investopedia / Michela Buttignol

Econometrics analyzes data using statistical methods in order to test or develop economic theory. These methods rely on statistical inferences to quantify and analyze economic theories by leveraging tools such as frequency distributions , probability, and probability distributions , statistical inference, correlation analysis, simple and multiple regression analysis, simultaneous equations models, and time series methods.

Econometrics was pioneered by Lawrence Klein , Ragnar Frisch, and Simon Kuznets . All three won the Nobel Prize in economics for their contributions. Today, it is used regularly among academics as well as practitioners such as Wall Street traders and analysts.

An example of the application of econometrics is to study the income effect using observable data. An economist may hypothesize that as a person increases their income, their spending will also increase.

If the data show that such an association is present, a regression analysis can then be conducted to understand the strength of the relationship between income and consumption and whether or not that relationship is statistically significant—that is, it appears to be unlikely that it is due to chance alone.

Methods of Econometrics

The first step to econometric methodology is to obtain and analyze a set of data and define a specific hypothesis that explains the nature and shape of the set. This data may be, for example, the historical prices for a stock index, observations collected from a survey of consumer finances, or unemployment and inflation rates in different countries.

If you are interested in the relationship between the annual price change of the S&P 500 and the unemployment rate, you'd collect both sets of data. Then, you might test the idea that higher unemployment leads to lower stock market prices. In this example, stock market price would be the dependent variable and the unemployment rate is the independent or explanatory variable.

The most common relationship is linear, meaning that any change in the explanatory variable will have a positive correlation with the dependent variable. This relationship could be explored with a simple regression model, which amounts to generating a best-fit line between the two sets of data and then testing to see how far each data point is, on average, from that line.

Note that you can have several explanatory variables in your analysis—for example, changes to GDP and inflation in addition to unemployment in explaining stock market prices. When more than one explanatory variable is used, it is referred to as multiple linear regression . This is the most commonly used tool in econometrics.

Some economists, including John Maynard Keynes , have criticized econometricians for their over-reliance on statistical correlations in lieu of economic thinking.

Different Regression Models

There are several different regression models that are optimized depending on the nature of the data being analyzed and the type of question being asked. The most common example is the ordinary least squares (OLS) regression, which can be conducted on several types of cross-sectional or time-series data. If you're interested in a binary (yes-no) outcome—for instance, how likely you are to be fired from a job based on your productivity—you might use a logistic regression or a probit model. Today, econometricians have hundreds of models at their disposal.

Econometrics is now conducted using statistical analysis software packages designed for these purposes, such as STATA, SPSS, or R. These software packages can also easily test for statistical significance to determine the likelihood that correlations might arise by chance. R-squared , t-tests ,  p-values , and null-hypothesis testing are all methods used by econometricians to evaluate the validity of their model results.

Limitations of Econometrics

Econometrics is sometimes criticized for relying too heavily on the interpretation of raw data without linking it to established economic theory or looking for causal mechanisms. It is crucial that the findings revealed in the data are able to be adequately explained by a theory, even if that means developing your own theory of the underlying processes.

Regression analysis also does not prove causation, and just because two data sets show an association, it may be spurious. For example, drowning deaths in swimming pools increase with GDP. Does a growing economy cause people to drown? This is unlikely, but perhaps more people buy pools when the economy is booming. Econometrics is largely concerned with correlation analysis, and it is important to remember that correlation does not equal causation.

What Are Estimators in Econometrics?

An estimator is a statistic that is used to estimate some fact or measurement about a larger population. Estimators are frequently used in situations where it is not practical to measure the entire population. For example, it is not possible to measure the exact employment rate at any specific time, but it is possible to estimate unemployment based on a randomly-chosen sample of the population.

What Is Autocorrelation in Econometrics?

Autocorrelation measures the relationships between a single variable at different time periods. For this reason, it is sometimes called lagged correlation or serial correlation, since it is used to measure how the past value of a certain variable might predict future values of the same variable. Autocorrelation is a useful tool for traders, especially in technical analysis.

What Is Endogeneity in Econometrics?

An endogenous variable is a variable that is influenced by changes in another variable. Due to the complexity of economic systems, it is difficult to determine all of the subtle relationships between different factors, and some variables may be partially endogenous and partially exogenous. In econometric studies, the researchers must be careful to account for the possibility that the error term may be partially correlated with other variables.

Econometrics is a popular discipline that integrates statistical tools and modeling for economic data, and it is frequently used by policymakers to forecast the result of policy changes. Like with other statistical tools, there are many possibilities for error when econometric tools are used carelessly. Econometricians must be careful to justify their conclusions with sound reasoning as well as statistical inferences.

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In the field of econometrics, hypothesis testing is an essential and widely used tool for making decisions and drawing conclusions based on data. The two main types of hypotheses in this field are the null hypothesis and the alternative hypothesis . These hypotheses play a crucial role in determining the validity of statistical results and ultimately, the accuracy of our conclusions. In this article, we will delve deeper into the concept of null and alternative hypotheses in econometrics, exploring their definitions, functions, and importance in hypothesis testing.

Whether you are a student studying econometrics or a professional in the field, understanding these fundamental concepts is crucial for conducting accurate and meaningful analyses. So, let's dive in and gain a comprehensive understanding of null and alternative hypotheses in econometrics. To start off, let's define what we mean by Null and Alternative Hypotheses . In simple terms, these are two opposing statements that are being tested in a statistical hypothesis test. The Null Hypothesis (H0) is the default position that there is no significant difference between groups or variables, while the Alternative Hypothesis (HA) is the opposite - it suggests that there is a significant difference. Now that we have a basic understanding of these hypotheses, let's explore how they are used in econometrics .

Econometrics is a branch of economics that combines economic theory, mathematics, and statistical analysis to study economic phenomena. These hypotheses play a crucial role in econometric research as they help us make conclusions about relationships between economic variables. In order to test these hypotheses, various methods and models are used in econometrics. These include regression analysis, time series analysis, panel data analysis , and more. Each method has its own set of assumptions and is used to answer specific research questions. Apart from methods, software and tools also play a significant role in econometrics.

Popular software such as STATA, EViews, and R are commonly used for data analysis in this field. These tools allow researchers to manipulate data, perform statistical tests, and create visual representations of the data to aid in their analysis. To better understand the role of Null and Alternative Hypotheses in econometrics, let's take a closer look at an example. Imagine a researcher wants to study the relationship between income and education level. The Null Hypothesis would state that there is no significant difference between income levels of individuals with different education levels.

The Role of Null and Alternative Hypotheses in Econometrics

This involves formulating a hypothesis, or a statement about a population parameter, and then using data to determine whether there is enough evidence to support or reject this hypothesis. In this process, the Null Hypothesis represents the default position, while the Alternative Hypothesis is the opposite of the Null Hypothesis. Understanding the basic principles of these hypotheses is crucial in econometrics as they serve as the foundation for statistical inference. The Null Hypothesis is usually denoted as H0 , while the Alternative Hypothesis is represented as Ha . These hypotheses play a significant role in determining the validity of econometric models and in making informed decisions based on statistical evidence. Moreover, understanding the concept of Type I and Type II errors is essential when working with Null and Alternative Hypotheses.

Type I error occurs when we reject a true Null Hypothesis, while Type II error happens when we fail to reject a false Null Hypothesis. Being aware of these potential errors can help researchers avoid drawing incorrect conclusions from their data. In conclusion, having a clear understanding of Null and Alternative Hypotheses is crucial in the field of econometrics. These hypotheses serve as the starting point for hypothesis testing and play a vital role in statistical inference. As you continue to explore the world of econometrics, remember the importance of these fundamental concepts and their role in shaping the validity of your research. In conclusion, Null and Alternative Hypotheses are fundamental concepts in econometrics that allow researchers to test their assumptions and make conclusions about economic relationships.

By using various methods, models, and software, we can gain a better understanding of economic phenomena and make informed decisions based on data-driven evidence.

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Introduction to Econometrics with R

7.3 joint hypothesis testing using the f-statistic.

The estimated model is

\[ \widehat{TestScore} = \underset{(15.21)}{649.58} -\underset{(0.48)}{0.29} \times STR - \underset{(0.04)}{0.66} \times english + \underset{(1.41)}{3.87} \times expenditure. \]

Now, can we reject the hypothesis that the coefficient on \(size\) and the coefficient on \(expenditure\) are zero? To answer this, we have to resort to joint hypothesis tests. A joint hypothesis imposes restrictions on multiple regression coefficients. This is different from conducting individual \(t\) -tests where a restriction is imposed on a single coefficient. Chapter 7.2 of the book explains why testing hypotheses about the model coefficients one at a time is different from testing them jointly.

The homoskedasticity-only \(F\) -Statistic is given by

\[ F = \frac{(SSR_{\text{restricted}} - SSR_{\text{unrestricted}})/q}{SSR_{\text{unrestricted}} / (n-k-1)} \]

with \(SSR_{restricted}\) being the sum of squared residuals from the restricted regression, i.e., the regression where we impose the restriction. \(SSR_{unrestricted}\) is the sum of squared residuals from the full model, \(q\) is the number of restrictions under the null and \(k\) is the number of regressors in the unrestricted regression.

It is fairly easy to conduct \(F\) -tests in R . We can use the function linearHypothesis() contained in the package car .

The output reveals that the \(F\) -statistic for this joint hypothesis test is about \(8.01\) and the corresponding \(p\) -value is \(0.0004\) . Thus, we can reject the null hypothesis that both coefficients are zero at any level of significance commonly used in practice.

A heteroskedasticity-robust version of this \(F\) -test (which leads to the same conclusion) can be conducted as follows:

The standard output of a model summary also reports an \(F\) -statistic and the corresponding \(p\) -value. The null hypothesis belonging to this \(F\) -test is that all of the population coefficients in the model except for the intercept are zero, so the hypotheses are \[H_0: \beta_1=0, \ \beta_2 =0, \ \beta_3 =0 \quad \text{vs.} \quad H_1: \beta_j \neq 0 \ \text{for at least one} \ j=1,2,3.\]

This is also called the overall regression \(F\) -statistic and the null hypothesis is obviously different from testing if only \(\beta_1\) and \(\beta_3\) are zero.

We now check whether the \(F\) -statistic belonging to the \(p\) -value listed in the model’s summary coincides with the result reported by linearHypothesis() .

The entry value is the overall \(F\) -statistics and it equals the result of linearHypothesis() . The \(F\) -test rejects the null hypothesis that the model has no power in explaining test scores. It is important to know that the \(F\) -statistic reported by summary is not robust to heteroskedasticity.

Introductory Econometrics

Chapter 17: joint hypothesis testing.

Chapter 16 shows how to test a hypothesis about a single slope parameter in a regression equation. This chapter explains how to test hypotheses about more than one of the parameters in a multiple regression model. Simultaneous multiple parameter hypothesis testing generally requires constructing a test statistic that measures the difference in fit between two versions of the same model.

An Example of a Test Involving More than One Parameter

One of the central tasks in economics is explaining savings behavior. National savings rates vary considerably across countries, and the United States has been at the low end in recent decades. Most studies of savings behavior by economists look at strictly economic determinants of savings. Differences in national savings rates, however, seem to reflect more than just differences in the economic environment. In a study of individual savings behavior, Carroll et al. (1999) examined the hypothesis that cultural factors play a role. Specifically, they asked the question, Does national origin help to explain differences in savings rate across a group of immigrants to the United States? Using 1980 and 1990 U.S. Census data with data on immigrants from 16 countries and on native-born Americans, Carroll et al. estimated a model similar to the following :( 1 )

For reasons that will become obvious, we call this the unrestricted model. The dependent variable is the household savings rate. Age and education measure, respectively, the age and education of the household head (both in years). The error term reflects omitted variables that affect savings rates as well as the influence of luck. The subscript h indexes households. A series of 16 dummy variables indicate the national origin of the immigrants; for example, Chinah = 1 if both husband and wife in household h were Chinese immigrants .( 2 ) Suppose that the value for the coefficient multiplying China is 0.12. This would indicate that, with other factors controlled, immigrants of Chinese origin have a savings rate 12 percentage points higher than the base case (which in this regression consists of people who were born in the United States).

If there are no cultural effects on savings, then all the coefficients multiplying the dummy variables for national origin ought to be equal to each other. In other words, if culture does not matter, national origin ought not to affect savings rates ceteris paribus. This is a null hypothesis involving 16 parameters and 16 equal signs:

The alternative hypothesis simply negates the null hypothesis, meaning that immigrants from at least one country have different savings rates than immigrants from other countries:

Now, if the null hypothesis is true, then an alternative, simpler model describes the data generation process:

Relative to the original model, the one above is a restricted model. We can test the null hypothesis with a new test statistic, the F-statistic, which essentially measures the difference between the fit of the original and restricted models above. The test is known as an F-test. The F-statistic will not have a normal distribution. Under the often-made assumption that the error terms are normally distributed, when the null is true, the test statistic follows an F distribution, which accounts for the name of the statistic. We will need to learn about the F- and the related chi-square distributions in order to calculate the P-value for the F-test.

F-Test Basics

The F-distribution is named after Ronald A. Fisher, a leading statistician of the first half of the twentieth century. This chapter demonstrates that the F distribution is a ratio of two chi-square random variables and that, as the number of observations increases, the F-distribution comes to resemble the chi-square distribution. Karl Pearson popularized the chi-square distribution beginning in 1900.

The Whole Model F-Test (discussed in Section 17.2) is commonly used as a test of the overall significance of the included independent variables in a regression model. In fact, it is so often used that Excel’s LINEST function and most other statistical software report this statistic. We will show that there are many other F-tests that facilitate tests of a variety of competing models. The idea that there are competing models opens the door to a difficult question: How do we decide which model is the right one? One way to answer this question is with an F-test. At first glance, one might consider measures of fit such as R2 or the sum of squared residuals (SSR) as a guide. But these statistics have a serious weakness – as you include additional independent variables, the R2 and SSR are guaranteed (practically speaking) to improve. Thus, naive reliance on these measures of fit leads to kitchen sink regression – that is, we throw in as many variables as we can find (the proverbial kitchen sink) in an effort to optimize the fit.

The problem with kitchen sink regression is that, for a particular sample, it will yield a higher R2 or lower SSR than a regression with fewer X variables, but the true model may be the one with the smaller number of X variables. This will be shown via a concrete example in Section 17.5. The F-test provides a way to discriminate between alternative models. It recognizes that there will be differences in measures of fit when one model is compared with another, but it requires that the loss of fit be substantial enough to reject the reduced model.

Organization

In general, the F-test can be used to test any restriction on the parameters in the equation. The idea of a restricted regression is fundamental to the logic of the F-test, and thus it is discussed in detail in the next section. Because the F-distribution is actually the ratio of two chi-square (?2) distributed random variables (divided by their respective degrees of freedom), Section 17.3 explains the chi-square distribution and points out that, when the errors are normally distributed, the sum of squared residuals is a random variable with a chi-square distribution. Section 17.4 demonstrates that the ratio of two chi-square distributed random variables is an F-distributed random variable. The remaining sections of this chapter put the F-statistic into practice. Section 17.5 does so in the context of Galileo’s model of acceleration, whereas Section 17.6 considers an example involving food stamps. We use the food stamp example to show that, when the restriction involves a single equals sign, one can rewrite the original model to make it possible to employ a t-test instead of an F-test. The t- and F-tests yield equivalent results in such cases. We apply the F-test to a real-world example in Section 17.7. Finally, Section 17.8 discusses multicollinearity and the distinction between confi- dence intervals for a single parameter and confidence regions for multiple parameters.

1 Their actual model is, not surprisingly, substantially more complicated. Return to text. 2 There were 17 countries of origin in the study, including 900 households selected at random from the United States. Only married couples from the same country of origin were included in the sample. Other restrictions were that the household head must have been older than 35 and younger than 50 in 1980. Return to text.

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Chapter 6 - hypothesis testing and confidence intervals.

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Learning Objectives

• Test a hypothesis about a regression coefficient 
• Form a confidence interval around a regression coefficient
• Show how the central limit theorem allows econometricians to ignore assumption CR4 in large samples 
• Present results from a regression model
• Central Limit Theorem in Action

What We Learned

• Our result is the same whether we drop CR4 and invoke the central limit theorem (valid in large samples) or whether we impose CR4 (necessary in small samples).
• Confidence intervals are narrow when the sum of squared errors is small, the sample is large, or there’s a lot of variation in X .
• How to present results from a regression model.

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Null Hypothesis

Null Hypothesis , often denoted as H 0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring. The null is t he truth or falsity of an idea in analysis.

In this article, we will discuss the null hypothesis in detail, along with some solved examples and questions on the null hypothesis.

Table of Content

What is Null Hypothesis?

Null hypothesis symbol, formula of null hypothesis, types of null hypothesis, null hypothesis examples, principle of null hypothesis, how do you find null hypothesis, null hypothesis in statistics, null hypothesis and alternative hypothesis, null hypothesis and alternative hypothesis examples, null hypothesis – practice problems.

Null Hypothesis in statistical analysis suggests the absence of statistical significance within a specific set of observed data. Hypothesis testing, using sample data, evaluates the validity of this hypothesis. Commonly denoted as H 0 or simply “null,” it plays an important role in quantitative analysis, examining theories related to markets, investment strategies, or economies to determine their validity.

Null Hypothesis Meaning

Null Hypothesis represents a default position, often suggesting no effect or difference, against which researchers compare their experimental results. The Null Hypothesis, often denoted as H 0 asserts a default assumption in statistical analysis. It posits no significant difference or effect, serving as a baseline for comparison in hypothesis testing.

The null Hypothesis is represented as H 0 , the Null Hypothesis symbolizes the absence of a measurable effect or difference in the variables under examination.

Certainly, a simple example would be asserting that the mean score of a group is equal to a specified value like stating that the average IQ of a population is 100.

The Null Hypothesis is typically formulated as a statement of equality or absence of a specific parameter in the population being studied. It provides a clear and testable prediction for comparison with the alternative hypothesis. The formulation of the Null Hypothesis typically follows a concise structure, stating the equality or absence of a specific parameter in the population.

Mean Comparison (Two-sample t-test)

H 0 : μ 1 = μ 2

This asserts that there is no significant difference between the means of two populations or groups.

Proportion Comparison

H 0 : p 1 − p 2 = 0

This suggests no significant difference in proportions between two populations or conditions.

Equality in Variance (F-test in ANOVA)

H 0 : σ 1 = σ 2

This states that there’s no significant difference in variances between groups or populations.

Independence (Chi-square Test of Independence):

H 0 : Variables are independent

This asserts that there’s no association or relationship between categorical variables.

Null Hypotheses vary including simple and composite forms, each tailored to the complexity of the research question. Understanding these types is pivotal for effective hypothesis testing.

Equality Null Hypothesis (Simple Null Hypothesis)

The Equality Null Hypothesis, also known as the Simple Null Hypothesis, is a fundamental concept in statistical hypothesis testing that assumes no difference, effect or relationship between groups, conditions or populations being compared.

Non-Inferiority Null Hypothesis

In some studies, the focus might be on demonstrating that a new treatment or method is not significantly worse than the standard or existing one.

Superiority Null Hypothesis

The concept of a superiority null hypothesis comes into play when a study aims to demonstrate that a new treatment, method, or intervention is significantly better than an existing or standard one.

Independence Null Hypothesis

In certain statistical tests, such as chi-square tests for independence, the null hypothesis assumes no association or independence between categorical variables.

Homogeneity Null Hypothesis

In tests like ANOVA (Analysis of Variance), the null hypothesis suggests that there’s no difference in population means across different groups.

• Medicine: Null Hypothesis: “No significant difference exists in blood pressure levels between patients given the experimental drug versus those given a placebo.”
• Education: Null Hypothesis: “There’s no significant variation in test scores between students using a new teaching method and those using traditional teaching.”
• Economics: Null Hypothesis: “There’s no significant change in consumer spending pre- and post-implementation of a new taxation policy.”
• Environmental Science: Null Hypothesis: “There’s no substantial difference in pollution levels before and after a water treatment plant’s establishment.”

The principle of the null hypothesis is a fundamental concept in statistical hypothesis testing. It involves making an assumption about the population parameter or the absence of an effect or relationship between variables.

In essence, the null hypothesis (H 0 ) proposes that there is no significant difference, effect, or relationship between variables. It serves as a starting point or a default assumption that there is no real change, no effect or no difference between groups or conditions.

The null hypothesis is usually formulated to be tested against an alternative hypothesis (H 1 or H [Tex]\alpha [/Tex] ) which suggests that there is an effect, difference or relationship present in the population.

Null Hypothesis Rejection

Rejecting the Null Hypothesis occurs when statistical evidence suggests a significant departure from the assumed baseline. It implies that there is enough evidence to support the alternative hypothesis, indicating a meaningful effect or difference. Null Hypothesis rejection occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

Identifying the Null Hypothesis involves defining the status quotient, asserting no effect and formulating a statement suitable for statistical analysis.

When is Null Hypothesis Rejected?

The Null Hypothesis is rejected when statistical tests indicate a significant departure from the expected outcome, leading to the consideration of alternative hypotheses. It occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

In statistical hypothesis testing, researchers begin by stating the null hypothesis, often based on theoretical considerations or previous research. The null hypothesis is then tested against an alternative hypothesis (Ha), which represents the researcher’s claim or the hypothesis they seek to support.

The process of hypothesis testing involves collecting sample data and using statistical methods to assess the likelihood of observing the data if the null hypothesis were true. This assessment is typically done by calculating a test statistic, which measures the difference between the observed data and what would be expected under the null hypothesis.

In the realm of hypothesis testing, the null hypothesis (H 0 ) and alternative hypothesis (H₁ or Ha) play critical roles. The null hypothesis generally assumes no difference, effect, or relationship between variables, suggesting that any observed change or effect is due to random chance. Its counterpart, the alternative hypothesis, asserts the presence of a significant difference, effect, or relationship between variables, challenging the null hypothesis. These hypotheses are formulated based on the research question and guide statistical analyses.

Difference Between Null Hypothesis and Alternative Hypothesis

The null hypothesis (H 0 ) serves as the baseline assumption in statistical testing, suggesting no significant effect, relationship, or difference within the data. It often proposes that any observed change or correlation is merely due to chance or random variation. Conversely, the alternative hypothesis (H 1 or Ha) contradicts the null hypothesis, positing the existence of a genuine effect, relationship or difference in the data. It represents the researcher’s intended focus, seeking to provide evidence against the null hypothesis and support for a specific outcome or theory. These hypotheses form the crux of hypothesis testing, guiding the assessment of data to draw conclusions about the population being studied.

Let’s envision a scenario where a researcher aims to examine the impact of a new medication on reducing blood pressure among patients. In this context:

Null Hypothesis (H 0 ): “The new medication does not produce a significant effect in reducing blood pressure levels among patients.”

Alternative Hypothesis (H 1 or Ha): “The new medication yields a significant effect in reducing blood pressure levels among patients.”

The null hypothesis implies that any observed alterations in blood pressure subsequent to the medication’s administration are a result of random fluctuations rather than a consequence of the medication itself. Conversely, the alternative hypothesis contends that the medication does indeed generate a meaningful alteration in blood pressure levels, distinct from what might naturally occur or by random chance.

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Example 1: A researcher claims that the average time students spend on homework is 2 hours per night.

Null Hypothesis (H 0 ): The average time students spend on homework is equal to 2 hours per night. Data: A random sample of 30 students has an average homework time of 1.8 hours with a standard deviation of 0.5 hours. Test Statistic and Decision: Using a t-test, if the calculated t-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: Based on the statistical analysis, we fail to reject the null hypothesis, suggesting that there is not enough evidence to dispute the claim of the average homework time being 2 hours per night.

Example 2: A company asserts that the error rate in its production process is less than 1%.

Null Hypothesis (H 0 ): The error rate in the production process is 1% or higher. Data: A sample of 500 products shows an error rate of 0.8%. Test Statistic and Decision: Using a z-test, if the calculated z-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: The statistical analysis supports rejecting the null hypothesis, indicating that there is enough evidence to dispute the company’s claim of an error rate of 1% or higher.

Q1. A researcher claims that the average time spent by students on homework is less than 2 hours per day. Formulate the null hypothesis for this claim?

Q2. A manufacturing company states that their new machine produces widgets with a defect rate of less than 5%. Write the null hypothesis to test this claim?

Q3. An educational institute believes that their online course completion rate is at least 60%. Develop the null hypothesis to validate this assertion?

Q4. A restaurant claims that the waiting time for customers during peak hours is not more than 15 minutes. Formulate the null hypothesis for this claim?

Q5. A study suggests that the mean weight loss after following a specific diet plan for a month is more than 8 pounds. Construct the null hypothesis to evaluate this statement?

Summary – Null Hypothesis and Alternative Hypothesis

The null hypothesis (H 0 ) and alternative hypothesis (H a ) are fundamental concepts in statistical hypothesis testing. The null hypothesis represents the default assumption, stating that there is no significant effect, difference, or relationship between variables. It serves as the baseline against which the alternative hypothesis is tested. In contrast, the alternative hypothesis represents the researcher’s hypothesis or the claim to be tested, suggesting that there is a significant effect, difference, or relationship between variables. The relationship between the null and alternative hypotheses is such that they are complementary, and statistical tests are conducted to determine whether the evidence from the data is strong enough to reject the null hypothesis in favor of the alternative hypothesis. This decision is based on the strength of the evidence and the chosen level of significance. Ultimately, the choice between the null and alternative hypotheses depends on the specific research question and the direction of the effect being investigated.

FAQs on Null Hypothesis

What does null hypothesis stands for.

The null hypothesis, denoted as H 0 ​, is a fundamental concept in statistics used for hypothesis testing. It represents the statement that there is no effect or no difference, and it is the hypothesis that the researcher typically aims to provide evidence against.

How to Form a Null Hypothesis?

A null hypothesis is formed based on the assumption that there is no significant difference or effect between the groups being compared or no association between variables being tested. It often involves stating that there is no relationship, no change, or no effect in the population being studied.

When Do we reject the Null Hypothesis?

In statistical hypothesis testing, if the p-value (the probability of obtaining the observed results) is lower than the chosen significance level (commonly 0.05), we reject the null hypothesis. This suggests that the data provides enough evidence to refute the assumption made in the null hypothesis.

What is a Null Hypothesis in Research?

In research, the null hypothesis represents the default assumption or position that there is no significant difference or effect. Researchers often try to test this hypothesis by collecting data and performing statistical analyses to see if the observed results contradict the assumption.

What Are Alternative and Null Hypotheses?

The null hypothesis (H0) is the default assumption that there is no significant difference or effect. The alternative hypothesis (H1 or Ha) is the opposite, suggesting there is a significant difference, effect or relationship.

What Does it Mean to Reject the Null Hypothesis?

Rejecting the null hypothesis implies that there is enough evidence in the data to support the alternative hypothesis. In simpler terms, it suggests that there might be a significant difference, effect or relationship between the groups or variables being studied.

How to Find Null Hypothesis?

Formulating a null hypothesis often involves considering the research question and assuming that no difference or effect exists. It should be a statement that can be tested through data collection and statistical analysis, typically stating no relationship or no change between variables or groups.

How is Null Hypothesis denoted?

The null hypothesis is commonly symbolized as H 0 in statistical notation.

What is the Purpose of the Null hypothesis in Statistical Analysis?

The null hypothesis serves as a starting point for hypothesis testing, enabling researchers to assess if there’s enough evidence to reject it in favor of an alternative hypothesis.

What happens if we Reject the Null hypothesis?

Rejecting the null hypothesis implies that there is sufficient evidence to support an alternative hypothesis, suggesting a significant effect or relationship between variables.

What are Test for Null Hypothesis?

Various statistical tests, such as t-tests or chi-square tests, are employed to evaluate the validity of the Null Hypothesis in different scenarios.

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OLS diagnostics: Model specification

This tutorial builds on the first four econometrics tutorials . It is suggested that you complete those tutorials prior to starting this one.

This tutorial demonstrates how to test for influential data after OLS regression. After completing this tutorial, you should be able to :

• Test model specification using the link test.
• Test for missing variables using the Ramsey regression specification error test (RESET).

Introduction

A common source of model specification error in OLS regressions is the omission of relevant variables. When variables are omitted, variations in the dependent variable may be falsely attributed to the included variables. This can result in inflated errors for regressors and can distort the estimated coefficients. In this tutorial, we will test for omitted variables using the link test and the Ramsey RESET test. Following previous tutorials, we've estimated an OLS model and stored the results using data simulates from the data generating process, $ y_{i} = 1.3 + 5.7 x_{i} + \epsilon_{i} $, where $ \epsilon_{i} $ is the random disturbance term.

The Link Test

The motivation behind the link test is the idea that if a regression is specified appropriately you should not be able to find additional independent variables. To test this, the link test regresses the dependent variable of the original regression against the original regression's prediction and the squared prediction. If the squared prediction regressor in the test regression is significant, there is evidence the model is misspecified.

To run the link test we construct the $\hat{y}$ and $\hat{y}^2$ variables from the results of the original regression and run the regression

$$y = \hat{y}b_1 + \hat{y}^2b_2 + \epsilon$$

The above code will print the following report:

The OLS results show a 53.7% p-value for our coefficient on $\hat{y}^2$. This suggests that we cannot reject the null hypothesis that the coefficient is equal to zero. This finding that the $\hat{y}^2$ is insignificant in our test regression suggests that our model does not suffer from omitted variables.

The Ramsey RESET Test

The Ramsey RESET test is based on the same concept but runs the regression

$$ y_i = x_ib + z_it + u_i $$

where $z_i = (\hat{y}^2, \hat{y}^3, \hat{y}^4$). The predicted $y$ value is normalized between 0 and 1 before the powers are calculated. If the regression is properly specified, the coefficients on all powers of the predicted $y$ should be jointly insignificant.

Normalize $\hat{y}$

To run the link test we need to normalize the predicted $y$ values, then construct the additional variables $\hat{y}^3$ and $\hat{y}^4$. To normalize the predicted $y$ from 0 to 1 we use min max normalization such that

$$y_{norm} = \frac{\hat{y}-\hat{y}_{min}}{\hat{y}_{max} - \hat{y}_{min}}$$

RESET regression

Unlike the link test, the Ramsey RESET test regression includes the regressors from the original regression:

$$y = xb_1 + \hat{y}^2b_2 + \hat{y}^3b_3 + \hat{y}^4b_4 + \epsilon$$

This time we will store the results because we need to conduct the hypothesis test that $b_2$, $b_3$, and $b_4$ are jointly insignificant.

The code above will print the following report:

RESET hypothesis test

To complete our RESET test for omitted variables we need to test the hypothesis that the coefficients on all powers of y_hat_norm are jointly insignificant. Therefore, the Ramsey RESET test null hypothesis is:

$$ H_0 : b_2 = b_3 = b_4 = 0 $$

using the F-statistics

$$F_0 = \frac{(SSR_r - SSR_{ur})/q}{SSR_{ur}/(n-(k+1))}$$

where $$SSR_r = \text{sum of squares restricted model}$$ $$SSR_{ur} = \text{sum of squares unrestricted model}$$ $$q = \text{number of restrictions}$$ $$n = \text{number of observations}$$ $$k = \text{number of regressors in unrestricted model}$$

In this case, the restricted model is $y = \alpha + \beta*x$, which is conveniently what we estimated in our original model.

The p-value for our F-stat is 10.4%. Therefore, at 5% significance level, we fail to reject the Ramsey RESET test null hypothesis of correct specification. This indicates that the functional form is correct and our model does not suffer from omitted variables.

Congratulations! You have:

• Calculated the link test model misspecification.
• Calculated the RESET test for model misspecification.

For convenience, the full program text is below.

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Null hypothesis stated as "there is an effect"

It is common to see $H_0$ stated as the inexistence of a difference between two data groups. The rejection of this hypothesis is always in favor of the alternative one that there is a difference. This makes a lot of sense, beginning from the fact that the name "null" suggests exactly this. However, I have the doubt: what would be the complications of stating $H_0$ as there being an effect? Could it be used to gather evidence in favor of the alternative hypothesis that the difference is 0?

I have a sense of why this would the answer to the latter question would be "no" (starting from the fact that we cannot prove a negative), but I'm looking for a more rooted response.

• hypothesis-testing
• econometrics
• equivalence

• 1 $\begingroup$ You can have a simple null hypothesis that the effect is $4$ or whatever. Or a composite null hypothesis that the effect is $\le 4$, or one with effect $\ge 4$ or one with the effect in the interval $[-4,4]$. What would usually make no sense is a null hypothesis that the effect is $\not=0$ if it would be difficult or impossible to reject such a hypothesis $\endgroup$ –  Henry Commented Nov 9, 2021 at 22:27
• $\begingroup$ Does this answer your question? While I addressed correlation, it could be any other parameter. // Note however that the null does not have to be zero. Your null could be $H_0: \theta = 4$, for example $\endgroup$ –  Dave Commented Nov 9, 2021 at 22:32
• $\begingroup$ A number of posts on site address this question. $\endgroup$ –  Glen_b Commented Nov 10, 2021 at 2:08
• $\begingroup$ Some other similar posts, of the many: stats.stackexchange.com/questions/264614/… , stats.stackexchange.com/questions/179527/… , stats.stackexchange.com/questions/444298/… $\endgroup$ –  kjetil b halvorsen ♦ Commented Nov 10, 2021 at 14:53
• $\begingroup$ The op's question is a great one. If we are testing a screw in a bridge for safety, when a faulty screw would mean that the portrait of the president who inaugurated the bridge would fall to the sea eventually, proceed in the usual way, i.e., we want to test whether the screw is faulty assuming that it is not. But if the failure of the screw implies that the bridge would collapse, then the null has to be that the screw is faulty, not the reverse. And likewise since there is so much clinical verbiage these days, a new procedure/treatment must be tested as if it were 'bad', and its goodness must $\endgroup$ –  Leonardo Auslender Commented Nov 16, 2021 at 23:26

You are, perhaps, thinking about a 'negativist' null hypothesis of the general form $\text{H}_{0}^{-}\text{: }|\theta|\ge \Delta$ , with $\text{H}_{\text{A}}^{-}\text{: }|\theta|< \Delta$ , as used in tests for equivalence, and where $\Delta$ is the smallest effect size that you care about a priori to the test. (As expressed here, the equivalence region ( $-\Delta, \Delta$ ) is symmetric, although it need not be. For example, one could express an asymmetric equivalence range $\text{H}^{-}_{0}\text{: }\theta \le \Delta_{\text{lower}}\textbf{ OR } \theta \ge \Delta_{\text{upper}}$ , where $|\Delta_{\text{lower}}| \ne \Delta_{\text{upper}}$ . One place this is sometimes done is when $\theta$ measures a relative difference like an odds ratio or relative risk, and where $\Delta_{\text{lower}} = \frac{1}{\Delta_{\text{upper}}}$ .)

In plain language, for a one-sample or two-sample test the null is "There is an effect of magnitude at least as large as $\Delta$ , and the alternative is "The magnitude of the effect is less that $\Delta$ ." The omnibus form of this null hypothesis would be "There is an effect of magnitude at least $\Delta$ between every group."

Note : For a continuous distribution we cannot simply invert the null hypothesis from the two-sided test for difference (i.e. $\text{H}_{0}^{+}\text{: }\theta = 0$ ) to be $\text{H}_{0}^{+}\text{: }\theta \ne 0$ , because we would have to find (probabilistic) evidence in favor of $\text{H}_{\text{A}}\text{: }\theta = 0$ , but the probability of a continuously distributed variable (e.g., normal, $t$ , etc.) exactly equaling a specific number is 0 (i.e. $P(X = c) = 0$ ), and so you would never reject such an inverted null (i.e. $\text{H}_{0}\text{: }\theta \ne 0$ is no good).

Selected References

Anderson, S., & Hauck, W. W. (1983). A new procedure for testing equivalence in comparative bioavailability and other clinical trials . Communications in Statistics—Theory and Methods , 12(23), 2663–2692.

Reagle, D. P., & Vinod, H. D. (2003). Inference for negativist theory using numerically computed rejection regions . Computational Statistics & Data Analysis , 42(3), 491–512.

Wellek, S. (2010). Testing Statistical Hypotheses of Equivalence and Noninferiority (Second Edition). Chapman and Hall/CRC Press.

See also: [ equivalence ], equivalence test , [ tost ]

• 1 $\begingroup$ This type of testing is also used in futility analyses in clinical drug development. Rather than looking for strong evidence to reject a null hypothesis of no treatment effect (or low end-of-study power), we retain a null hypothesis that the treatment does work (or there is sufficient end-of-study power) based on weak evidence to the contrary. $\endgroup$ –  Geoffrey Johnson Commented Nov 9, 2021 at 22:57
• $\begingroup$ @GeoffreyJohnson Yup! A really cool heritage of clinical/pharmacological biostats. Propagating to other statistical sciences now as well... seeing it in psychology, I published with it in Demography , and PLoS ONE , and teach my biostats students to help avoid confirmation bias by combining inferences from both tests for difference and tests for equivalence (i.e. to form 'relevance test' conclusions). $\endgroup$ –  Alexis Commented Nov 9, 2021 at 23:00
• $\begingroup$ Say, @GeoffreyJohnson I wonder if you might be a good person to have a go at answering my question Is there a generalized concept of noncentrality of a distribution? ? There are some answers there which are helpful, but none really land it for me (I have Fisher's original derivation of the non-central $t$ on hold with ILL… need to go pick that up… maybe it will help :). $\endgroup$ –  Alexis Commented Nov 9, 2021 at 23:14
• 1 $\begingroup$ Thank you, Alexis! Pretty clear. $\endgroup$ –  Santiago Valdivieso Commented Nov 18, 2021 at 20:11

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The test of significance approach

Assume the regression equation is given by yt = a + 3xt + ut, t = 1, 2,..., T. The steps involved in doing a test of significance are shown in box 2.5.

Box 2.5 Conducting a test of significance

(1) Estimate a, ¡3 and SE(a), SE0) in the usual way.

(2) Calculate the test statistic. This is given by the formula test statistic = ——3— (2.30)

where 3* is the value of 3 under the null hypothesis. The null hypothesis is H0 : 3 = 3* and the alternative hypothesis is H1 : 3 = 3* (for a two-sided test).

(3) A tabulated distribution with which to compare the estimated test statistics is required. Test statistics derived in this way can be shown to follow a t-distribution with T — 2 degrees of freedom.

(4) Choose a 'significance level', often denoted a (not the same as the regression intercept coefficient). It is conventional to use a significance level of 5%.

(5) Given a significance level, a rejection region and non-rejection region can be determined. If a 5% significance level is employed, this means that 5% of the total distribution (5% of the area under the curve) will be in the rejection region. That rejection region can either be split in half (for a two-sided test) or it can all fall on one side of the y-axis, as is the case for a one-sided test.

For a two-sided test, the 5% rejection region is split equally between the two tails, as shown in figure 2.13.

For a one-sided test, the 5% rejection region is located solely in one tail of the distribution, as shown in figures 2.14 and 2.15, for a test where the alternative is of the 'less than' form, and where the alternative is of the 'greater than' form, respectively.

Figure 2.13

Rejection regions for a two-sided 5% hypothesis test f(x)

Figure 2.14

Rejection region for a one-sided hypothesis test of the form

Figure 2.15

Rejection region for a one-sided hypothesis test of the form #0:3 = 3*, #1:3 > 3*

2.5% rejection region

Rejection regions for a two-sided 5% hypothesis test

Box 2.5 contd.

(6) Use the t-tables to obtain a critical value or values with which to compare the test statistic. The critical value will be that value of x that puts 5% into the rejection region.

(7) Finally perform the test. If the test statistic lies in the rejection region then reject the null hypothesis (Ho), else do not reject H0.

Steps 2-7 require further comment. In step 2, the estimated value of p is compared with the value that is subject to test under the null hypothesis, but this difference is 'normalised' or scaled by the standard error of the coefficient estimate. The standard error is a measure of how confident one is in the coefficient estimate obtained in the first stage. If a standard error is small, the value of the test statistic will be large relative to the case where the standard error is large. For a small standard error, it would not require the estimated and hypothesised values to be far away from one another for the null hypothesis to be rejected. Dividing by the standard error also ensures that, under the five CLRM assumptions, the test statistic follows a tabulated distribution.

In this context, the number of degrees of freedom can be interpreted as the number of pieces of additional information beyond the minimum requirement. If two parameters are estimated (a and p - the intercept and the slope of the line, respectively), a minimum of two observations is required to fit this line to the data. As the number of degrees of freedom increases, the critical values in the tables decrease in absolute terms, since less caution is required and one can be more confident that the results are appropriate.

The significance level is also sometimes called the size of the test (note that this is completely different from the size of the sample) and it determines the region where the null hypothesis under test will be rejected or not rejected. Remember that the distributions in figures 2.13-2.15 are for a random variable. Purely by chance, a random variable will take on extreme values (either large and positive values or large and negative values) occasionally. More specifically, a significance level of 5% means that a result as extreme as this or more extreme would be expected only 5% of the time as a consequence of chance alone. To give one illustration, if the 5% critical value for a one-sided test is 1.68, this implies that the test statistic would be expected to be greater than this only 5% of the time by chance alone. There is nothing magical about the test -- all that is done is to specify an arbitrary cutoff value for the test statistic that determines whether the null hypothesis would be rejected or not. It is conventional to use a 5% size of test, but 10% and 1% are also commonly used.

However, one potential problem with the use of a fixed (e.g. 5%) size of test is that if the sample size is sufficiently large, any null hypothesis can be rejected. This is particularly worrisome in finance, where tens of thousands of observations or more are often available. What happens is that the standard errors reduce as the sample size increases, thus leading to an increase in the value of all t-test statistics. This problem is frequently overlooked in empirical work, but some econometricians have suggested that a lower size of test (e.g. 1%) should be used for large samples (see, for example, Leamer, 1978, for a discussion of these issues).

Note also the use of terminology in connection with hypothesis tests: it is said that the null hypothesis is either rejected or not rejected. It is incorrect to state that if the null hypothesis is not rejected, it is 'accepted' (although this error is frequently made in practice), and it is never said that the alternative hypothesis is accepted or rejected. One reason why it is not sensible to say that the null hypothesis is 'accepted' is that it is impossible to know whether the null is actually true or not! In any given situation, many null hypotheses will not be rejected. For example, suppose that H0 : = 0.5 and H0 : = 1 are separately tested against the relevant two-sided alternatives and neither null is rejected. Clearly then it would not make sense to say that 'H0 : 3 = 0.5 is accepted' and 'H0 : 3 = 1 is accepted', since the true (but unknown) value of cannot be both 0.5 and 1. So, to summarise, the null hypothesis is either rejected or not rejected on the basis of the available evidence.

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Readers' Questions

How to test for parameter significance in econometrics?

There are several statistical tests that can be used to test for parameter significance in econometrics. Here are three commonly used tests: t-test: The t-test is used to determine if the estimated coefficient for a particular parameter is significantly different from zero. The null hypothesis is that the estimated coefficient is equal to zero, indicating no relationship between the independent variable and the dependent variable. If the t-statistic (calculated by dividing the estimated coefficient by its standard error) is greater than a critical value at a chosen significance level (e.g., 0.05 or 0.01), then the null hypothesis is rejected, suggesting that the parameter is statistically significant. F-test: The F-test is used to test the joint significance of multiple parameters. It is typically applied when testing the significance of multiple independent variables in a multiple regression model. The null hypothesis is that all the coefficients being tested are equal to zero. If the calculated F-statistic is greater than the critical value at a chosen significance level, then the null hypothesis is rejected, indicating that at least one of the tested parameters is statistically significant. Wald test: The Wald test is another method to test the significance of individual parameters in a regression model. It is based on the chi-square distribution and can be used for both linear and nonlinear models. The null hypothesis is that the parameter is equal to a specified value (usually zero). If the calculated Wald statistic exceeds the critical value at a chosen significance level, the null hypothesis is rejected, indicating parameter significance. It is important to note that these tests have different assumptions and application requirements, so it is essential to choose the appropriate test based on the specific econometric model and research objectives. Additionally, it is always recommended to consult with relevant statistical software or econometrics textbooks for proper testing procedures.

What is rejection region?

A rejection region is a predetermined interval on a statistical test, such as a z-test, t-test, or chi-squared test, in which data values that fall outside of the interval signify that the null hypothesis has been rejected. Rejection regions are used in hypothesis testing to decide whether to accept or reject a hypothesis.

How to run a twosided tail test for econometrics?

A two-sided tail test for econometrics is used to test for the equality of parameters (i.e., their mean or median) across different populations or two different subgroups. This is accomplished by computing the statistic for a two-sided test and then comparing it to a critical value from a two-sided critical value table. The two-sided critical value table is the same as the one used for a one-sided tail test, but the differences are that the degrees of freedom and the alpha level used to determine the critical value may be different. The statistic used for a two-sided test is typically a t-statistic or a z-statistic, depending on the number of observations available in the data set. To run a two-sided tail test for econometrics, you will need to first calculate the statistic for the two-sided test. Next, you will need to compare this statistic to the critical value from the two-sided tail table with the appropriate degrees of freedom and alpha level. If the calculated statistic is larger than the critical value, then this indicates that there is a statistically significant difference between the two populations or two subgroups. If the calculated statistic is smaller than the critical value, then this indicates that there is not a statistically significant difference between the two populations or two subgroups.

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A four laws structure for looking at economics through the eyes of thermodynamics.

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Costa, V.A.F. A Four Laws Structure for Looking at Economics through the Eyes of Thermodynamics. Energies 2024 , 17 , 4223. https://doi.org/10.3390/en17174223

Costa VAF. A Four Laws Structure for Looking at Economics through the Eyes of Thermodynamics. Energies . 2024; 17(17):4223. https://doi.org/10.3390/en17174223

Costa, Vítor A. F. 2024. "A Four Laws Structure for Looking at Economics through the Eyes of Thermodynamics" Energies 17, no. 17: 4223. https://doi.org/10.3390/en17174223

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