(c) i) In ation (INFL) is regressed on the unemployment rate (UNR):

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1 BRUNEL UNIVERSITY Master of Science Degree examination Test Exam Paper EC500: Modelling Financial Decisions and Markets EC5030: Introduction to Quantitative methods Model Answers. COMPULSORY (a) xy x :75 (b) The r from the regression of y on x is the variation in y that is explained (linearly) by x. r ESS TSS TSS-RSS RSS TSS TSS : (c) i) In ation (INFL) is regressed on the unemployment rate (UNR): INFLa + bunr + u: ii) A lagged response is not unreasonable since time is required for unemployment to a ect wages and further time for wage changes to lter through to the prices of goods: iii) One can t the reciprocal relation: INFLa + bunr(-) + u: INFLa + b UNR(-) + u: iv) Finally, one can t (by nonlinear least squares) the nonlinear relation: INFLa + b UNR(-)- + u: (d) The speci cation of the linear model centers on the disturbance vector u and the matrix. The assumptions are as follows: with y + u; E(u) 0; E(uu 0 ) I: In other words, u i, i ; : : : ; n are iid N(0; ). column rank k. Moreover, is nonstochastic with full

2 (e) The Box-Pierce statistic is based on the squares of the rst p autocorrelation coef- cients of the OLS residuals. The statistic is de ned as where r j Q n p n j r j ; tj+ e te t n Under the hypothesis of zero autocorrelations for the residuals, Q will have an asymptotic distribution, with degrees of freedom equal to p minus the number of parameters estimated in tting the ARMA model. An improved small-sample performance is expected from the revised Ljung-Box statistic Q 0 n(n + ) p t e t j j r j (n j): : (f) The hypothesis is k + l with test statistic t b k + b l p var(bk + b l ) b k + b l p var(bk ) + var(b l ) + cov(b k ; b l ) 0:63 + 0:45 p 0:57 + 0:9 + 0:055 0:78: In order to get the critical values we need the sample size. This is not signi cant for any sample size. We cannot reject the hypothesis of constant returns to scale.. (a) See Lecture Notes. (b) In all cases the null hypothesis is rejected. Therefore, there is evidence of: heteroscedasticity; serial correlation in the residuals and squared residuals; Non-normality. Finally, according to Ramsey s reset test the null hypothesis: H 0 rejected. : u N(0; I) is 3. (a) If the unknown vector is replaced by some guess or estimate b, this de nes a vector of residuals e, e Y b ; (n) (n) (nk)(k) The least-squares principle is to choose b to minimize the residual sum of squares, n e 0 e; e 0 [e e e n ]: e 0 e (Y b) 0 (Y b) (Y 0 b 0 0 )(Y b) Y 0 Y b 0 0 Y Y 0 b + b 0 0 b: i e i Next, note that Y 0 b is a scalar and hence its transpose (Y 0 b) 0 b 0 0 Y is the (n)(nk)(k) same scalar (Y 0 b). Thus, b 0 0 Y Y 0 b.

3 From the above analysis it follows that The rst order conditions are e 0 e Y 0 0 b 0 0 Y + b 0 0 b: 0 Y + 0 b 0; giving the least-squares b vector as a function of the data: 0 b 0 Y; or b ( 0 ) 0 Y. (b) Under rational expectations, the regression r + r + u; should yield a 0 and b. We can carry out this test by the F-test in part a. In this case: 0 0 R ; r : 0 The test statistic is F (b r)0 ( 0 )(b r)q s :4 0: :568 0:4 0:06 :998: The 5% critical value is The test statistic is not signi cant and we cannot reject the rational expectations hypothesis. 4. (a) i) Assume that k 3. Then R (qk) b 4 b (k) b b b 3 3 r : (q) Example : The null hypothesis is: H 0 : b b 3 or b b 3 0 (q ). Example : The null hypothesis is: H 0 : b b 3 0 (q ). For the foregoing examples we have: 5 : ) R 0, r 0; ) R ; r : ii) The vector of the restrictions is: Rb-r. 3

4 The estimated variance covariance matrix of the coe cient vector (b) is: var(b) s ( 0 ). Thus, var(rb) var(rb r) s R( 0 ) R 0. Note that ( 0 ) is a (k k) matrix, R is a (q k) matrix and R 0 is a (k q) matrix. In other words var(rb r) is a (q q) matrix. A computable statistic, which has an F-distribution under the null hypothesis is (Rb r) 0 [var(rb r)] (Rb r)q F (q; n k): or (Rb r) 0 [s R( 0 ) R 0 ] (Rb r)q F (q; n k): Note that (Rb r) is a (q) column vector, (Rb r) 0 is a (q) row vector and [s R( 0 ) R 0 ] is a (q q) matrix. In other words the computable statistic is a scalar. iii) The test procedure is then to reject the null hypothesis Rb r if the computed F value exceeds a preselected critical value. (b) The likelihood function is given by Therefore, the log-likelihood function is L(; x) Y n f(x i) i Y n xi e i n n e x i i The rst order condition is which gives the MLE l n b n n n i x i n i x i: i x i 0; x : (c) In the two-variable equation we have: Var(b) Var(a) Cov(a; b) Cov(a; b) Var(b) ( 0 ) : 0 in this case is " 0 # N : Thus, ( 0 ) " # N N ( ) : 4

5 Then, we have Var(b) N N ( ) ( ) N N ( ) : 5. (a) One simply computes an auxiliary regression of the squared OLS residuals on a constant and all nonredundant variables in the set consisting of the regressors, their squares, and their cross products. Suppose, for example, that by t a + b x t + b x t ; and e t y t by t. Then, the auxiliary regression is e t on ; x t ; x t ; x t; x t; and x t x t. On the hypothesis of homoscedasticity, nr is asymptotically distributed as (5). The degrees of freedom are the number of variables in the auxiliary regression (excluding the constant). In general, under the null of homoscedasticity, nr a (q); where q is the number of variables in the auxiliary regression less than one. If homoscedasticty is rejected, there is no indication of the form of the heteroscedasticity and, thus no guide to an appropriate estimator. Computing the White standard errors would be, however, wise if one proceeds with OLS. (b) Recall that the least-squares vector b is related to the data b ( 0 ) 0 Y; where Substituting for Y gives Y + u: b ( 0 ) 0 ( + u) + ( 0 ) 0 u: () In taking expectations we have E(b). Conventional OLS coe cient standard errors are incorrect, and the conventional test statistics based on them are invalid. The correct variance matrix for the OLS coe cient is Var(b) E[( 0 ) 0 uu 0 ( 0 ) ] ( 0 ) 0 E(uu 0 )( 0 ) ( 0 ) 0 ( 0 ) : () The conventional formula calculates ( 0 ), which is only part of the correct expression in (). 5

6 The White estimator replaces the unknown t (t ; : : : ; n) by e t, where the e t denote the OLS residuals. This provides a consistent estimator of the variance matrix for the OLS coe cient vector and is particularly useful because it does not require any speci c assumptions about the form of the heteroscedasticity. The empirical implementation of Eq. () is then Estimated Var(b) ( 0 ) 0 ( b 0 ) ; where b diagfe ; : : : ; e ng: The square roots of the elements on the principal diagonal of estimated Var(b) are the estimated standard errors of the OLS coe cients. They are often referred to as heteroscedasticityconsistent standard errors. (c) Ramsey has argued that the various sepici cation errors (omitted variables, incorrect functional form, correlation between and u) give rise to a nonzero u vector. Thus the null and alternative hypotheses are H 0 : u N(0; I); H : u N(; I): The test of H 0 is based on the augmented regression y + Za + u: The test for the speci cation error is then a 0. ramsey s suggestion is that Z shold contain powers of the predicted values of the dependent variable. For example: where by b, and by [by by : : : by n]: Z [by by 3 by 4 ] 6