ECONOMETRICS EC331 Prof. Burak Saltoglu Statistical Inference

Topic to be covered Simple versus Multiple regression Interpretation of coefficients estimates Hypothesis testing: testing the significance of the model as a whole

Testing the Overall Significance of a Regression With k-Variable regression if we want to test the hypothesis

Testing the Overall Significance of a Regression We have the test statistic, k: number of parameters α: level of significance n: sample size

Testing the Overall Significance of a Regression On Eviews estimating three variable case

Testing the Overall Significance of a Regression Results as Checking individual significance The F statistic p value

Testing the Overall Significance of a Regression An alternative way:

Testing the Overall Significance of a Regression Then, with to test the hypothesis You can calculate the same F as

Testing the Overall Significance of a Regression On Eviews Result Screen

Testing the Marginal Contribution of a Variable How can we measure marginal effects of explanatory variables on explanatory power of the model? or alternatively If you are using the R2 version, make sure dependent variables in two models identical, otherwise use the former.

Testing the Marginal Contribution of a Variable X2 variable statistically a significant variable to be added or not

Testing the Marginal Contribution of a Variable NEW Regression result is

Testing the Marginal Contribution of a Variable , So adding X2 to our model, significantly increases model’s total explanatory power,

Testing Linear Restrictions on Parameters Suppose we given the Cobb-Douglas production function Taking logarithms So we now have a linear model.How do we test the null of constant returns to scale

Testing Linear Restrictions on Parameters 1) t-Test Approach: Test Statistic will be If t exceeds critical t value, reject the null of This method is easy to implement, but via this method, you investigate a hypothesis after the estimation. Somehow we need to embed the restriction in to estimation process.

Testing Linear Restrictions on Parameters 2) Restricted Least Squares: We can build a new regression upon the restriction we have Plugging in is our restricted regression

Testing Linear Restrictions on Parameters Now, we have unrestricted(UR) and restricted(R) regressions. We have the test statistic or n: sample size k: number of parameters in UR regression q: the number of restrictions

Testing Linear Restrictions on Parameters Let’s do an example with EViews.(another version of example 8.3) The UR regression is We are using RSS version, because dependent variables are different in two regressions

Testing Linear Restrictions on Parameters Restricted regression estimation result

Testing Linear Restrictions on Parameters We have So since F statistic<F critical we fail to reject null of constant returns to scale. That is, there is evidence in favour of constant returns to scale in this economy

Chow Test Chow tests can be conducted to see whether there is a structural change during the sample period. Suppose you are working with data between 1970-1995 and there is the suspect of a structural change in 1982. If there is no structural change, you estimate for whole sample range, 1970-1995

Chow Test But if there is a structural change 1982 and parameters of model has changed, the correct specification must be for 1970-1981 for 1982-1995

Chow Test The procedure as follows: 1) Estimate whole sample with n1+n2 observations.Note that you are applying restrictions Take the RSS and name it restricted RSS, RSSR 2) Estimate two sub-samples, having n1 and n2 observations, respectively, individually and get RSS1 and RSS2 then

Chow Test 3) And test statistic is But assumptions of Chow test are u1 and u2 are independently distributed—no serial correlation Both u1 and u2 are distributed by N(0,σ2) – homoscedasticity

Chow Test Let’s perform an example with Eviews.Suppose we are given linearized money demand function(real values in logarithms): and have quarterly data between 1960-2003. Suppose, we suspect of a breakdown in the first quarter of 1971 when Nixon declared that USA abandoned Gold Standard. So we have two sub samples: 1- 1960 to end of 1970 2- 1971 to end of 2003

First fit the whole sample with 176(=n1+n2) observations Chow Test First fit the whole sample with 176(=n1+n2) observations

Chow Test Then fit 1960-1970 with n1=44 observations

Chow Test and fit 1971 to 2003 with n2=132 observations

Chow Test Then RSS for unrestricted regression will be F statistic is F statistic exceeds the critical value, so we reject the null hypothesis, so there is evidence of parameter breakdown.

Dummy Variables

Outline Why Dummty Use of Intercept Dummy Variables Intercept Dummies Slope Dummy Variables Comparing Two Regressions with Dummy Variables (Test for Structural Change)

Introduction Dummy Variables are explanatory variables that take one of two values, usually 0 or 1. Dummy Variables are useful for capturing the qualitative characteristics like gender, race or geographic region. Interaction: Age and gender,

Use of Intercept Dummy Variables intercept may change for some of the observations in the sample. Example: US consumption and income during 1929-1970.

Use of Intercept Dummy Variables How did consumption change before and after WWII. Therefore the relation during war times and reconstruction period. consider the parameter changes during war. So we should develop a way to incorporate the qualitative factors to our model. A general a dummy variable is;

Intercept Dummy In our example; Then our model is; Therefore the estimated consumption is;

Use of Intercept Dummy Variables

Intercept Dummy In Matrix Notation

Slope Dummy Variables The new variable is the product of income and the dummy variable is called interaction variable. Alternatively it can be called as a slope dummy variable.

Slope Dummy Variables C Y

Slope Dummy In Matrix Notation

Comparing Two Regressions with Dummy Variables As we have already discussed the Chow test is the way to test structural changes in the relationship. Same can be tested by using dummy variables. Let us consider our example of consumption and income relationship, if there is a structural change in our data, the dummy variables in the model must be statistically significant. If they’re significant this shows that there has been structural change in consumption and income relationship during WWII.

Comparing Two Regressions with Dummy Variables First regress the following model; Obtain t values for and If the t values exceed the critical value, this means there is structural change otherwise there is not.

Some Related Topics Regime Switching Models and Business Cycles Seasonal Effects (daily, montly hourly dummies) Test for structural changes: New regime in variance New regime in skewness New regime in kurtosis and even higher moments Dummies and Autocorrelation

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End of the Lecture