Esame Applied Econometrics Banking and Finance PDF

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Applied Econometrics

1 out of ten

Applied Econometrics November 24, 2019

Andrea Monticini

Part A 06 marks Are the following statements true or false? Explain your answers in a maximum of one or two sentences each. 1) The total area under any probability density function is less than 1. 2) The probability that a drawing from the N (0, 2) distribution is a number less than 2 is 0.5. 3) When choosing between alternative regression models, one should be guided by selecting the model with the largest R2 . 4) In the regression model yi = α + βxi + ui , the least squares estimator of β imposing α = 0 has the formula Pn yi xi Pi=1 . n 2 i=1 xi 5) The error of rejecting a true null hypothesis is called a Type I error. Part B 6 marks Multiple choice questions. Answer all the following questions by writing a, b, c, or d in your answer book. There is only one correct answer (2 marks each). 1) In the regression model yi = α + βxi + ui , to perform a test at 5% level of the hypothesis H0 : β = 0 against the alternative H1 : β 6= 0, assuming the sample is large one should a) Use the chi-squared distribution. ˆ ˆ exceeds 1.64. b) Reject if the ratio β/s.e.( β) ˆ ˆ exceeds 1.96. c) Reject if the ratio | β/s.e.( β)| d) Reject unless βˆ < 0 .

Applied Econometrics

2 out of ten

2) The F statistic for the test of restrictions in a regression model is a function of a) the ratio of the restricted and unrestricted sums of squares. b) the difference between the restricted and unrestricted sums of squares, divided by the restricted sum of squares. c) the difference between the restricted and unrestricted sums of squares, divided by the unrestricted sum of squares. d) the ratio of the restricted coefficient to its standard error. 3) In the regression model yi = β1 x1i + β2 x2i + β3 + ui suppose that x2i is omitted. In the equation yi = γ1 x1i + γ2 + vi which of the following is the correct definition of γ1 ? a) γ1 = β1 + β2 . b) γ1 = β1 + δβ2 where δ is the coefficient in the regression of x1i on x2i . c) γ1 = β1 + δβ2 where δ is the coefficient in the regression of x2i on x1i . d) γ1 = β1 . 4) Hypothesis testing in time series econometrics depends mainly on which assumption for its validity? a) The classical regression model assumptions hold. b) The data vary systematically. c) The sample is large. d) The t distribution is close to the normal distribution.

Applied Econometrics

3 out of ten

5) Consider the following model yt = β1 x1t + β2 x2t + β3 + ǫt we want to test H0 : β2 = +3β1 . Which of the following procedures should you follow? a) test γ2 = 0 in the regression yt = γ1 x1t + γ2 zt + γ3 + ǫt , where zt = x1t + 3x2t ; b) test γ2 = 0 in the regression yt = γ1 zt + γ2 x2t + γ3 + ǫt , where zt = x1t − 3x2t ; c) test γ2 = 0 in the regression yt = γ1 x1t + γ2 zt + γ3 + ǫt , where zt = x1t − 3x2t ; d) test γ2 = 0 in the regression yt = γ1 zt + γ2 x2t + γ3 + ǫt , where zt = x1t + 3x2t . 6) Consider a linear regression model with k-regressors (including a constant), the biased Maximum Likelihood error variance estimator is a) b) c) d)

RSS n RSS n−k

q

RSS n

q

RSS n−k

Applied Econometrics

4 out of ten

Part C 6 marks Answer any ONE of these questions. 1) Suppose that the observed x can be represented as the sum of the true x˜ and a random measurement error ν, x = x˜ + ν The true relationship is y = βx + u − βν

(1)

a) After discussing the hypothesis behind the classical linear regresˆ sion model, derive β. ˆ b) Derive the plim β. c) Explain why an instrumental variable can help in estimating the equation (1) 2) Discuss three out of four of the following concepts a) Consistency b) Stationarity c) GARCH models d) Autocorrelation

Applied Econometrics

5 out of ten

Part D 18 marks Answer all these questions. Consider the following model △ut = δ + δ0 gt + δ1 gt−1 + θ1 △ut−1 + θ2 △ut−2 + εt

(2)

where △ut = ut − ut−1 , ut is the unemployment rate in period t, gt is the growth rate of output in period t and εt is a white noise. Based on the below R-output, answer ALL the following questions. a) Is the above model (2) well specified? In other words, would you include an additional variable gt−2 to the regression model (2)? b) Comments on the statistical significance of the estimated parameters. c) What do the reported diagnostic tests tell us about this regression? Describe briefly the problem that each test is designed to detect. d) Find the impact multiplier. e) Find the total multiplier using the following algorithm: α = δ/ (1 − θ1 − θ2 ) β 0 = δ0 β 1 = δ 1 + β 0 θ1 βj = βj−1 θ1 + βj−2 θ2 j > 2

f) Find the normal growth rate GN (hint: the normal growth rate that P is needed to maintain a constant rate of unemployment, GN = −α/ ∞ j=0 βj ).

Applied Econometrics

6 out of ten

Applied Econometrics

7 out of ten

Applied Econometrics

8 out of ten

Answer Part A: 1. F 2. F 3. F 4. T 5. T Part B: 1. c 2. c 3. c 4. c 5. d 6. a Part C: 1.a)-b) The assumptions are 1) 2) 3) 4)

The relationship is linear y = Xβ + u u ∼ i.i.d(0, σ 2 ) The regressors X are exogenous (or fixed) The regressors are linearly independent (i.e. it is possible to  −1 compute X T X )

b Now, we have to derive β.

y = βx + (u − βν)

(3)

which shows that if we model y as a function of x, the transformed disturbance contains the measurement error in x P x (u − βν) P 2 βb = β + (4) x

Applied Econometrics

9 out of ten

Then plim

1X 1X 1X x (u − βν) = plim xu − β plim xν = −βσν2 n n n

Substitution in eq (4) gives plim βb = β −

βσν2 =β σx˜2 + σν2



σx˜2 σx˜2 + σν2



(5)

c) What we need is an IV for x. Such an IV must be correlated with x, uncorrelated with u and uncorrelated with the measurement error, ν . 2)

a) The sample mean is then said to be a consistent estimator of µ when lim P r (| x¯n − µ |< ǫ) = 1 n→∞

or plim x¯n = µ b) “A time series x1 , x2 , x3 , ... is stationary if the joint distribution of a segment xt , ..., xt+m is the same as that of a segment xt+k , ...., xt+m+k for any choices of t, m and k”. {xt }is stationary (in strict sense) if, for every k > 0, the joint distributions of collections (xt , xt+1 , xt+2 , ..., xt+k ) do not depend in any way on t. {xt }is stationary (in wide sense) if the mean E[xt ], the variance Var(xt ), and the autocovariances Cov(xt , xt+j ) for j > 0 are all independent of t. c) In the specification y = Xβ + u, one hopes to include all relevant variables in the X matrix. In such a case the disturbances would be expected to be serially uncorrelated. However, we may have Pure autocorrelation: distribution of the error term has autocorrelation nature, a shock to GDP persists for more than one period Impure autocorrelation – Inadequate specification of the model: suppose the true model is yt = β0 + β1 xt + β2 yt−1 + ut i.i.d

and ut ∼ (0, σ 2 ). The researcher specifies yt = β0 + β1 xt + vt

Applied Econometrics

10 out of ten

the pseudodisturbance is then vt = β2 yt−1 + ut , which is autocorrelated because the correct specification makes y autocorrelated. – Inadequate specification of the error term: error-in-variables Part D: a) Based on the BIC there is no need to add gt−2 . b) All the estimated coefficients are statistically significant at 5% level. c)

At 5% level we cannot reject the null hypothesis of no-autocorrelation. At 1% level we reject the null hypothesis of a linear relationship At 1% level we reject the null hypothesis of NO heteroskedasticity

d) The impact multiplier is −0.09040 e) The total multiplier is −0.3917976 f) The normal growth rate is

−0.3229616 −0.3917976

= 0.8243073...