FENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK 4. Prof. Mei-Yuan Chen Spring 2008

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1 FENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK 4 Prof. Mei-Yuan Chen Spring 008. Partition and rearrange the matrix X as [x i X i ]. That is, X i is the matrix X excluding the column x i. Let u i denote the residual vector of regressing y on X i and v i denote the residual vector of regressing x i on X i. Define the partial correlation coefficient of y and x i as r i u i v i (u i u i) / (v i v i) / Let R i and R be the coefficients of determination obtained from the regressions of y on X i and y on X, respectively. (a) Applying matrix inversion formula to show I P (I P i ) (I P i )x i x i (I P i ) x i (I P i )x, i where P X(X X) X and P i X i (X i X i) X i. (b) Show that ( R )/( R i ) r i, Using this result to verify R R i r i ( R i ). What does this result tell you? (c) Let τ i denote the t-ratio of ˆβ it, the i-th element of ˆβ T obtained from regressing y on X. First show that τ i (T k)r i /( r i ). Using this result to verify r i τ i /(τ i + T k). (d) Combining the results in (b) and (c) to show R R i τ i ( R )/(T k). What does this result tell you?

2 . Let X,...,X T be independent random variables with the density function f(x;p) ( p) x p. Find the MLE for p. 3. Consider the model y i β 0 + β x i + ǫ i, i,,t, where x i are non-stochastic and ǫ i are independently distributed as N(kx i, σ0 ). Find the MLE for β 0 and β. 4. Given the model y t β +β x t +...+β k x tk +ǫ t, consider the standardized regression: y t β x t + + β k x tk + ǫ t, where β i are known as the beta coefficients, y t y t ȳ, x ti x ti x i s xi, ǫ t ǫ t ǫ, with s y (T ) t (y t ȳ) and s x i (T ) t (x ti x i ). (a) What are the relationships between β i and β i? Give an interpretation of the beta coefficients. (b) Are the t-ratios of the standardized regression different from those of the original regression? 5. Given the model y t β x t + β x t + β 3 x t3 + ǫ t with β +β α and β +β 3 α, suppose that all the classical assumptions hold. (a) As α is unknown, how do you test this constraint in the original model? (b) How would you estimate α? Iour estimator ˆα the BLUE? 6. Suppose that a linear model with k explanatory variables has been estimated. (a) Show that ˆσ T Centered TSS( R )/(T ). What does this result tell you?

3 (b) Suppose that we want to test the hypothesis that s coefficients are zero. Show that the F-test can be written as φ (T k + s)ˆσ c (T k)ˆσ u sˆσ u, where ˆσ c and ˆσ u are the variance estimates of the constrained and unconstrained models, respectively. By setting a (T k)/s also show that ˆσ c ˆσ u a + φ a +. (c) Based on the results in (a) and (b), what can you say if φ > or φ <? 3

4 ECONOMETRICS I Answer Key for Homework 4 Prof. Mei-Yuan Chen Apring 008. (a) Using matrix inversion formula in Greene (993, p. 7) and letting X [X i x i ], we get [ X (X X) i X i X i x ] i x i X i x i x i ( ) (X i X i) I + X i x ix i X i(x i X i) x i (I P i)x i (X i X i) X i x i x i (I P i)x i. x i X i(x i X i) x i (I P i)x i x i (I P i)x i When X [x i X i ] [ x (X X) i x i x i X i X i x i X i X i x i (I P i)x i (X i X i) X i x i It is then easy to verify that ] x i X i(x i X i) x i (I P i)x i ( x i (I P i)x i (X i X i ) I + X i x ix i X i(x i X i) x I P I [x i X i ][ i x i x i X i X i x i X i X i I P i (I P i)x i x i (I P i) x i (I P i )x. i ] [ x i X i (b) Note that u i (I P i )y and v i (I P i )X i. Then R R i y (I P)y y (I P i )y x i (I P i)x i ) y (I P i )y (y (I P i )x i ) /(x i (I P i )x i ) y (I P i )y r i. This result hold for both centered and non-centered R. (c) By Frisch-Waugh-Lovell Theorem, ˆβ it [x i (I P i)x i ] x i (I P i)y. By (a), var(ˆβ ˆ it ) σt x i (I P i )x i y (I P)y (T k)[x i (I P i )x i ] 4 ].

5 y (I P i )y (y (I P i )x i ) /(x i (I P i)x i ) (T k)[x i (I P i)x i ] y (I P i )yx i (I P i)x i (y (I P i )x i ) (T k)[x i (I P i)x i ] Therefore, since r i [y (I P i )x i ] /[y (I P i )yx i (I P i )x i ], ˆβ it τi var(ˆβ ˆ it ) ( ) x i (I P i )y ( (T k)[x i (I P i)x i ] ) x i (I P i)x i y (I P i )yx i (I P i)x i (y (I P i )x i ) ( [y (I P (T k) i )y][x i (I P i )x i ] ) [x i (I P i)y] T k /ri (d) Straightforward. (T k)r i r i. The MLE is p / x, where x T t x t. 3. The MLE are β i (x i x)(y i ȳ) i (x i x) k,. β 0 ȳ β x, where x T i x i and ȳ T i y i. These results should be obvious because the model can be written as y i β 0 + (β + k)x i + ǫ i, where ǫ i is distributed as N(0, σ 0 ). Hence, we can obtain the MLE for β 0 and β +k using standard formula. 4. As ȳ β + β x + + β k x k + ǫ, we have y t ȳ β x t x β s x }{{} β x t x s x }{{} x t x + + β kt x k k + ǫ t ǫ + + β k s xk 5 }{{} β k x kt x k s xk } {{ } x kt + ǫ t ǫ. }{{} ǫ t

6 This gives the standardized regression. When x it changes by one unit, i.e., x it changes by one standard deviation s xi, then yt will change by β i, i.e., y t will change by βi. This standardization thus permits comparison among regression coefficients. That the t-ratios remain the same can be easily verified using a simple linear regression model. 5. The constraints imply that β +β +β 3 0, hence a simple t-test on this hypothesis will do. Also observe that y t β x t + β x t + β 3 x 3t + ǫ t β x t + (α β )x t (α + β )x 3t + ǫ t β (x t x t x 3t ) + α(x t x 3t ) + ǫ t. The OLS estimator ˆα remains to be the BLUE because all the classical assumptions are still valid. 6. (a) The result follows from the fact that (b) R e e/(t k) (y y Tȳ )/(T ) ˆσ T TSS/(T ). Thus, R increases whenever ˆσ T decreases. φ (ESS c ESS u )/s ESS u /(T k) (e c e c e u e u )/s e u e u /(T k) (T k + s)ˆσ c (T k)ˆσ u sˆσ u. It is straightforward to show that for a (T k)/s, ˆσ c ˆσ u a + φ a +. (c) φ > implies ˆσ c > ˆσ u. Hence by (a), R c < R u. That is, when φ >, dropping these s variables would reduce R. Note that whether φ is significant does not matter. Similarly, φ < implies ˆσ c < ˆσ u, and hence R c > R u. 6

7 FENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK 3 Prof. Mei-Yuan Chen Spring 008. Given the model y Xβ 0 + e, where X is T k. Let ˆβ T denote the OLS estimator and R k denote the resulting centered R, where the subscript k signifies a model with k explanatory variables. (a) Show that R k k i ˆβ it t (x ti x i )y t t (y t ȳ), where ˆβ it is the i-th element of ˆβ T, x ti is the t-th element of the i-th explanatory variable, y t is the t-th element of y, x i t x ti /T, and ȳ t y t /T. (b) Suppose that you delete an explanatory variable from the model (so that the model has k explanatory variables) and obtain R k, show that R k R k.. Consider the model y Xβ 0 + e, where X does not contain the constant term. (a) Show that y y T(ȳ) ŷ ŷ T( ŷ) + ê ê T( ê) T ŷ ê. (b) If we use R Centered RSS Centered TSS, or ESS R Centered TSS, are they bounded between zero and one? How should one compute R in the model without a constant term? 3. Suppose that we estimate the model y Xβ 0 + e and obtain ˆβ T and centered R. (a) If y 000 y and X are used as the dependent and explanatory variables, what is the effect of this change on ˆβ T and R? (b) If y and X 000 X are used as the dependent and explanatory variables, what is the effect of this change on ˆβ T and R? (c) If y and X are used as the dependent and explanatory variables, what is the effect of this change on ˆβ T and R? 4. Find a condition under which R is negative. 7

8 ECONOMETRICS I Answer Key for Homework 3 Prof. Mei-Yuan Chen Spring 008. (a) Let x t be the t-th column of X, then R k t (ŷ t ȳ) t (y t ȳ) t [ˆβ T (x t x)]ŷ t t (y t ȳ) ˆβ t T (x t x)(ŷ t + ê t ) k t (y ˆβ i it t (x ti x i )y t t ȳ) T t (y. t ȳ) Note that this expression holds when the model has a constant term. (b) Using the result of (a), k (k) Rk i ˆβ T it t (x ti x i )y t t (y, t ȳ) R k k i t (x ti x i )y t t (y, t ȳ) (k ) ˆβ it ˆβ (k) T ˆβ (k ) where and T are the OLS estimates for models with k and k variables, respectively. Note that the first k elements of are different form ˆβ (k ) T ˆβ (k) T in general. Suppose that Rk > R k. Then the estimator ˇ β (k) T [ˆβ (k ) ˆβ (k ) ˆβ (k ) k 0] yields the coefficient of determination Rk for the model with k variables. This contradicts the LS principle of maximizing R.. (a) Since y y ŷ ŷ + ê ê and ȳ ŷ + ê, T(ȳ) T( ŷ) + T( ê) + T ŷ ê, we get the answer. That is, y y T(ȳ) ŷ ŷ T( ŷ) + ê ê T( ê) T ŷ ê. (b) When a model does not contain the constant term, the centered R need not be bounded between 0 and, and non-centered R should be used. Note that, R Centered RSS Centered TSS >, if ê ê T( ê) T ŷ ê < 0; R ESS Centered TSS < 0, if ŷ ŷ T( ŷ) T( ê) T ŷ ê < (a) ˆβ T 000 ˆβ T, and R is unchanged. (b) ˆβ T ˆβ T /000, and R is unchanged. (c) ˆβ T and R are unchanged. 4. R < 0 when R < (k )/(T k). 8

9 FENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK Prof. Mei-Yuan Chen Spring 008. Consider a location regression model with T observations {x,x,...,x T }: x t α + e t,t,...,t. (a) What is X as shown in the lecture note for this model? (b) What is the OLS estimator, ˆα T, for α? (c) What is the variance of the sampling distribution of ˆα T? (d) What is the sample variance you suggested for ˆα T?. Show algebraically the following results (a) ˆβ T (ryx)( t y )/( (b) ˆβ T (r yx )( t xy) T T t y /T)/( t x /T) where ˆβ T is the OLS estimate of the linear regression model y t α + βx t + u t,t,...,t. 3. A regresion model: y t βx t +u t,t, is considered. If u and u are statistically independent with common mean 0 and variance σu, find the sampling distribution of the following two estimators of the slope coefficient: t ˇβ y t t x, ˆβ t y tx t. t t x t Show that var(ˇβ) > var(ˆβ). 4. Given data on y and x, construct a linear regression model for each equation below and explain how you can estimate the parameters α and β. (a) y α + β log x. (b) y αx β. (c) y αe βx. x (d) y αx β. 9

10 (e) y eα+βx + e α+βx. 5. Consider the bivariate normal distributions is specified by f(x,y) Q(x, y) π σ exp[q(x,y)] y ρ { (x ) ( )( ) ( ) } µx x µx y µy y µy ( ρ ρ + ) σ y σ y (a) What is the conditional density function of Y on X x? (b) What is the conditional mean of Y on X x? (c) What is the conditional variance of Y on X x? 0

11 ECONOMETRICS I Answer Key for Homework Prof. Mei-Yuan Chen Spring 008. (a) x x. x T α. + e e. e T. (b) Q(α) t (x t α) /T and the first-order condition becomes Q(α) α T (x T t α) set 0.. (a) t Denote the solution as ˆα T which satisfies t (x t ˆα T ) 0, we have ˆα T t x t/t x T. (c) var(ˆα T ) var( x T ) σ X /T. (d) s ˆα T s x T s X /T, where s X t (x t ˆx T ) /(T ) is the sample variance of σ X. ˆβ n i (x i x n )(y i ȳ n ) i (x i x n ) [ i (x i x n )(y i ȳ n )] [ i (x i x n ) ][ i (y i ȳ n ) ] rxy i (y i ȳ n ) n i (x i x n )(y i ȳ n ). i (y i ȳ n ) i (x i x n )(y i ȳ n ) (b) ˆβ n i (x i x n )(y i ȳ n ) i (x i x n ) i (x i x n )(y i ȳ n ) i (x i x n ) i (y i ȳ n ) i (y i ȳ n ) i (x i x n ) r xy i (y i ȳ n ) i (x i x n ).

12 3. Since y t βx t + u t,t,, and we have var(ˇβ) σ u ( t x t ), var(ˆβ) var(ˇβ) var(ˆβ) It is easy to have t σ u t x t σu ( t x t) σ u t x t σ u[ t x t ( t x t) ] ( t x t). t x t x t ( x t ) (x + x ) (x + x ) x x x + x (x x ). We get the proof. t 4. (a) Regres on and log x. (b) ln y ln α + β ln x, regress ln y on and ln x to get lnˆ α and ˆβ. (c) ln y ln α + βx, regress ln y on and x to get lnˆ α and ˆβ. (d) (/y) α β(/x), regress /y on and (/x) to get ˆα and ˆβ. (e) y/( y) e α+βx, regress ln(y/( y)) on and x to get ˆα and ˆβ. 5. As Q(x, y) ( ρ ) ( ρ ) { (x ) ( )( µx y µy x µx ρ σ y { (y ) ( )( ) µy y µy x µx ρ σ y ( ) x ( ) } ρ µx x µx + ( ρ ) { (y µy σ y ρ x µ x σ y ) + ( ) } y µy σ y + ρ ( x µx ) ( ) } x + ( ρ µx ) ) Therefore, f(x,y) π σ y ρ exp[q(x,y)]

13 [ { (y exp µy πσy ρ ( ρ ρ x µ ) }] x ) σ y [ { ( ) }] x exp πσx ( ρ ( ρ µx ) ) ( ) exp y µ y ρσy (x µ x ) πσy ρ ( ρ )σy [ (x µx ) ] exp πσx f(y x)f(x). σ x It is ready to have that the conditional mean is µ y x µ y + ρσ y (x µ x ), and the conditional variance is var(y x) σ y ( ρ ). 3

14 FENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK Prof. Mei-Yuan Chen Spring 008. Suppose that random variables x and y take only two values 0 and, and have the following joint probability function x 0 x y y Find E(y x), E(y x) and var(y x) for x 0 and x.. The value of the mean of a random sample of size 0 from a normal population X is x n 8.. Find the 95 % confidence interval for the mean of the population on the assumption that the variance is σ X Let x n be the mean of a random sample of size n from an N(µ,σ ) population. What is the probability that the interval ( x n σ/ n, x n + σ/ n) includes the point µ? 4. The mean of a random sample of size 7 from a normal population is x n 4.7. Determine the 90 % confidence interval for the population mean when the estimate variance of the population is Suppose a simple linear regression model is considered for the conditional mean of Y on X x a t α + βx t + e t for a random sample {(y t,x t ),t,...,t }. (a) What is the functional form of the conditional mean implied by the supposed regression model? (b) Are the parameters of the function of conditional mean assumed to be constant over the whole sample? (c) What are the OLS estimators for α and β? (d) What are the variances of the OLS estimators for α and β and the covariance between them? 4