AGEC 621 Lecture 16 David Bessler
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1 AGEC 621 Lecture 16 David Bessler This is a RATS output for the dummy variable problem given in GHJ page 422; the beer expenditure lecture (last time). I do not expect you to know RATS but this will give you knowledge of what standard outputs look like and allow us to analyze the output and relationships between T-stats and F- tests. RATS Run on Oct (c) Thomas A. Doan. All rights reserved * Beer Data * 621 lecture 24 * *Ignore the next several lines, it s just RATS input stuff. cal alloc :1 eqv BE S E1 E2 E3 Y AGE E1S E2S E3S YS declare symmetric v open data a:beer.txt data(format=free,org=obs) 1901:1 1940:1 1 to 7 set E1S 1901:1 1940:1 = %x(t,2)*%x(t,3) set E2S 1901:1 1940:1 = %x(t,2)*%x(t,4) set E3S 1901:1 1940:1 = %x(t,2)*%x(t,5) set YS 1901:1 1940:1 = %x(t,2)*%x(t,6)
2 Here we ask the computer to do an ols estimation in RATS commands (the (vcv) matrix asks for the variance covariance matrix on coefficients we will use this later). linreg(vcv) :1 1940:1 # constant Dependent Variable BE - Estimation by Least Squares Annual Data From 1901:01 To 1940:01 Usable Observations 40 Degrees of Freedom 30 Centered R** R Bar ** Uncentered R** T x R** Mean of Dependent Variable Std Error of Dependent Variable Standard Error of Estimate Sum of Squared Residuals Regression F(9,30) Significance Level of F Durbin-Watson Statistic Q(10-0) Significance Level of Q Variable Coeff Std Error T-Stat Signif ********************************************************* 1. Constant S E E E Y E1S E2S E3S YS
3 Covariance\Correlation Matrix of Coefficients Constant S E1 E2 Constant S E E E Y E1S E2S E3S YS E3 Y E1S E2S Constant S E E E Y e E1S E2S E3S YS e E3S YS Constant S E E E Y E1S E2S E3S YS e-006
4 F-Tests on Restrictions First we want to compare the F-test (from restricted versus an unrestricted ols regressions) with the t-test as given on the previous page. restrict 1 # 3 # F(1,30)= with Significance Level Notice that my F(1,30) significance level of is the same as the significance level on the t-test associated with the E1 variable from the ols regression. Next I do the same thing for the coefficient associated with E2 restrict 1 # 4 # F(1,30)= with Significance Level Again check the significance level on E2 from the ols output. Finally I look at a linear combination of estimated coefficients. restrict 1 # 3 4 # F(1,30)= with Significance Level How do we do this last test (without just invoking the magic of the computer!)? This lecture focuses attention on hypothesis testing on a linear combination of coefficients.
5 Recall the formula for the variance of a linear combinations of variables. Say x 1, x 2, x 3,..., x k are k random variables. Then the variance of the sum of a linear combination of these variables is given as: Var (a 1 x 1 + a 2 x 2 + a 3 x a k x k ) k = G (a i 2 VAR (x i )) + G G a i. a j COV (x i,x j ) i=1 i j What does this formula for the variance of a linear combination of variables mean? For constants a 1, a 2,..., a k when combine the random variables x 1, x 2,..., x k by multiplying each by it s associated constant and add these, the sum (the result) has a variance given by the right-hand-side of the above equation. We square each of the constant weights (a i ) and multiply each by the variance of each of the associated random variables (x i ) we add these. To this sum we add the covariances between i and j (i not equal to j) weighted by the product of the constants (a i times a j ).
6 Let s write this out for k = 2: Var (a 1 x 1 + a 2 x 2 ) = (a 1 2 VAR (x 1 )) + (a 2 2 VAR (x 2 )) + a 1. a 2 COV (x 1,x 2 ) + a 2. a 1 COV (x 2, x 1 ) = (a 1 2 VAR (x 1 )) + (a 2 2 VAR (x 1 )) + 2 a 1. a 2 COV (x 1, x 2 ) Again but now for k = 3: Var (a 1 x 1 + a 2 x 2 + a 3 x 3 ) 2 = (a 1 VAR (x 1 )) + (a 2 2 VAR (x 2 ) + (a 2 3 VAR (x 3 )) + a 1. a 2 COV (x 1, x 2 ) + a 1. a 3 COV (x 1, x 3 ) + a 2. a 1 COV (x 2, x 1 ) + a 2.a 3 COV (x 2, x 3 ) + a 3. a 1 COV (x 3, x 1 ) + a 3, a 2 COV (x 3, x 2 ) Var (a 1 x 1 + a 2 x 2 + a 3 x 3 ) = (a 1 2 VAR (x 1 )) + (a 2 2 VAR (x 2 ) + (a 3 2 VAR (x 3 )) + 2 a 1. a 2 COV (x 1, x 2 ) + 2 a 1. a 3 COV (x 1, x 3 ) + 2 a 2. a 3 COV (x 2, x 3 ) Now what does all this have to do with hypothesis testing? Well, say we want to test the hypothesis that : $ 1 = $ 2. We could write this as: Ho: $ 1 - $ 2 = 0 We test this using the formulas we derived earlier:
7 t = (b 1 b 2 0) / (Var (b 1 b 2 )) ½ Now in the formula given above the x 1 is b 1, a 1 is 1, x 2 is b 2, and a 2 is 1 (k=2). Let s test the hypothesis that the coefficient associated with E1 equals the coefficient associated with E2 in the above problem: t= ( ) / ( ( )).5 = Comparing this with the t-table for T-K = 30 degrees of freedom at a 5% significance level we fail to reject the null hypothesis that the two betas are equal.
8 Make sure you can pick out the correct variances and covariances from the above printout. (On a test I ll give you the matrix just like above, so be sure you know what is going on - - correlations above the diagonal, covariances below the diagonal and variances on the diagonal.) Just a point of note, if we performed this test as an F-test, where we restricted versus unrestricted test, we would get a calculated u-value (see last lecture on u) of.00953, which at F(1,30) again suggests that we do not reject the null hypothesis. It turns out that if we take our calculated t value from above and square it ( ) we get (.00953). This result is general t 2 = F (for a single linear restriction, it s not clear on multiple restrictions even how to do the t-test). Let s try this out this last bit of insight on a single hypothesis test, say that the beta associated with E1 is equal to zero (in the regression out given above). Now from the regression output we see the t-value is ( / ). Now if we square we get.67132; which is the F-value computed above for the test that the beta on E1 = 0 (see the restricted test given above). Another Example Test the hypothesis that the coefficients associated with E1, E2 and E3 in the above output sum to 100. Here all of the a i weights are 1.0 as: Ho: $ 1 + $ 2 + $ = 0; don t worry that the estimated coefficient of the third beta is negative, the hypothesis ask only that we test the sum of the coefficients: t = ( ) / [ = , ( ) + 2( ) + 2( )].5
9 This last figure is what I get when I run a restricted versus unrestricted test using the F-statistic. For our purposes please be sure you can calculate the t-test on a linear combination of variables.
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