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Linear regressions.

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Introduction

Linear regressions

Problem 1

You have estimated the linear regression model

yt = a + b1x1t + b2x2t + b3x3t + et

using annual data for the period 1960-94. Explain briefly how you would construct a test of the model’s forecasting performance using additional data for the period 1995-98.

Solution.

First, estimate the model and obtain estimations for the coefficients a, b1, b2, b3 using 1960-94 data.

Then, obtain forecast for the 1995-98 based on estimated model together with errors of forecast. So, the intervals for the values for the 1995-98 will be obtained.

After this one can check whether actual data fit into the obtained intervals or not. Also, one can check how far are the actual values from the forecast.

Problem 2

Describe briefly how you would test whether the OLS residuals from the linear regression model

Yt = a + bXt + ut

are serially correlated. Outline how you would modify the specification of your model, or

the estimation procedure, if your test revealed showed significant serial correlation.

Solution

One may estimate the initial regression and obtain regression residuals:

image00.png

then one should estimate the regression of et on its lag:

et=ρet-1+zt

If the coefficient ρ appeared to be significant, then there is serial correlation in residuals.

...read more.

Middle

Data period 1991Q1-1997Q4 (28 observations) SSR = 0.0010334

(iii) Test the hypothesis that the coefficients of Model A are constant over the period 1979-1997 against the alternative that there is a structural break after 1990Q4.

Solution

Let us call constant in the regression as β1, coefficients on D4Ct-1 - D4Ct-4as β25, coefficients on D4Yt-1 - D4Yt-4  as β69 correspondingly.

(i)

The test is simply a Fisher test i.e. β23=…=β9=0.

One has to calculate F-statistic (according to model A):

image01.png

The critical value for F-statistic is F0.95(k-1,n-k)=F0.95(8,56)=2.1

Since 35>2.1 then null hypothesis on all the coefficient are zeroes has to be rejected, i.e. hypothesis that the four quarter change in the logarithm of consumption is unaffected by any lagged variables is rejected.

An economist is interested in this hypothesis to check whether consumption in current period is affected by previous periods or is determined by current income only.

(ii)

The tested hypothesis is H0:  β6=…=β9=0

To test the hypothesis one has to calculate Fisher statistic (model A is unrestricted regression, model B is restricted regression i.e. regression in which β6=…=β9=0):

image02.png

Critical value for the F-statistic is F0.95(q,n-k)=F0.95(4,56)=2.536

Since 1,69<2.526 then  the hypothesis is accepted

(iii)

...read more.

Conclusion

(a) A fellow-student, B, suggests to A that he can increase the size of his dataset, while avoiding the effort involved in collecting more data, just by duplicating each of his existing observations the appropriate number of times. Explain why a dataset which is extended in this way fails to satisfy the assumptions of the Gauss-Markov theorem, and identify the assumption which will not hold.

(b) Suppose A follows B’s advice, and constructs an extended dataset which consists of a number of copies of observation 1, followed by a number of copies of observation 2, and so on. Explain which standard diagnostic test is likely to reveal his deception.

<Think how residuals may reveal the problem>

Solution

(a)

The dataset obtained in this way fails to satisfy the assumptions of the Gauss-Markov theorem since errors et for duplicated observations are the same, so they are not independent, as required by Gauss-Markov Theorem. To be more precise, the condition that E[εtεs]=0 when t ≠s is not satisfied for s and t are numbers of the pair of duplicated observations

(b) In this case errors are correlated with order 1 (and probably there is higher order correlation). So, the deception is to be revealed by Darbin-Watson test which will show high positive autocorrelation in errors (i.e. DW will be close to 4).

...read more.

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