REGRESSION AND COEFFICIENT

QUESTION

This is a theoretical and applied econometrics project to help you familiarize yourself with the recent materials covered in class and an econometric software of your choice.

You must answer all questions. You should prepare a word-processed report as your answer. Give full details of results obtained and any inferences which have been drawn from them. Computer codes used could also be included as part of your results or in an appendix to the assignment (depending on the relevance and length of the programs). You must also include short excerpts of the computer output (or graphs) to justify your answers. The final report should not exceed 10 pages including printouts of estimation outputs, printouts of code and mathematical proofs. It should be submitted in PDF format

References

Campbell, J. Y. (2003). Consumption-based asset pricing. In Constantinides, G., Harris, M., and Stulz, R. M., editors, Handbook of the Economics of Finance, volume 1 of Handbook of the Economics of Finance, chapter 13, pages 803{887. Elsevier.

Yogo, M. (2004). Estimating the elasticity of intertemporal substitution when instruments are weak. The Review of Economics and Statistics, 86(3):797{810.

The elasticity of intertemporal substitution (EIS) in consumption is a parameter of prime importance in macroeconomics and _nance. In macroeconomics, it relates current and ex- pected future real interest rates to the current level of aggregate demand in the intertemporal IS equation. In _nance, in the consumption and portfolio choice problem of an in_nite-lived investor, the EIS is the key parameter in the optimal consumption rule. The EIS, denoted by  , is estimated from the following model

_ct+1 = _i +  ri;t+1 + ui;t+1 (1)

where _ct+1 is the consumption growth at time t + 1, ri;t+1 is the real return on asset i at t + 1, and _i is a constant. The error ui;t+1 is correlated with the regressor, the real return on asset. The interest is in ascertaining whether   < 1. When the EIS is less (greater) than 1, the investors optimal consumption-wealth ratio is increasing (decreasing) in expected returns. Further information can be found in Yogo (2004) and Campbell (2003). Estimate the EIS in equation (1) using two asset returns separately: the real interest rate (rf ) and the real aggregate stock return (re) for France and then for the UK (4 equations to estimate). The real stock return is constructed as log of the gross stock return deated by the consumer price index. The real interest rate is constructed in the same way, using an available proxy for the short-term interest rate. Real consumption growth is the _rst di_erence in log real consumption per capita. The following variables could also be used as potential instruments: nominal interest rate, ination, consumption growth, and log dividend-price ratio. Be careful in indexing time t of your instruments appropriately (ie: do you need to lag these instrument?). The data is available in \Data Assignment2.xls”. When answering your questions, state any assumptions you need to make and show all your workings. You are welcome to use di_erent software packages (e.g. Matlab, R, Gretl, Stata, etc.) for di_erent parts of the question if it is helpful to you.

(a) Estimate the model by OLS and report your results.

(b) Do you need instrumental variables? Select a set of instruments and justify your choice.

(c) Re-estimate the equation by generalized instrumental variables, treating the two asset returns as endogenous:

i. Are the estimates signi_cantly di_erent from the OLS ones? Perform a Hausman test for the hypothesis that the two asset returns can be treated as exogenous.

ii. Verify that the IV estimates may also be obtained by two-stages OLS regression. Are the reported standard errors the same? Explain why or why not.

(d) Test for the overidentifying restrictions for the IV estimation you performed. How do you interpret the results of this test?

(e) Are your instruments any good? Could you use other instruments or more instruments?

Perform any relevant test and re-estimate the equation if necessary.

(f) Test for heteroskedasticity and serial correlation. If your tests reject the null(s), correct for the problem(s) by computing the appropriate variance-covariance and the appropri- ate estimator. Justify the estimator your are using in comparison to the other estimator you used above. [Hint: the best answers to this question will use an iterative procedure.]

SOLUTION

ASSIGNMENT

PART A- OLS REGRESSION

PART B

Yes we do need the use of instrumental variables because the error term of the regression equation is correlated with the explanatory variables, which in this case is the real rate of return and the real aggregate stock return. This violates the OLS assumption that the covariance between the explanatory variables and the error term is equal to zero. Hence the OLS estimates are not consistent.

We can use nominal interest rate, inflation, consumption growth and log dividend price ratio as the IVs as these are correlated with the asset returns and uncorrelated with the error term as well as exogenous.

PART C- GENERALIZED INSTRUMENTAL VARIABLE METHOD

FRANCE

2SLS

!ST STAGE- regressing the real interest rate on all the instrumental variables

Regression of real aggregate stock return on all the instrumental variables

2SLS FOR U.K

1^ST step is to regress  rrf to all the instrumental variables

Regress rr to all the instrumental variables

Last step is to regress the dependent variable on the two predicted values.

HAUSMANN TEST

FRANCE

Comment- hence Ho= the variable rrf is exogenous is accepted because the p value is greater than .05 significant level.

Similarly for rr we have

COMMENT- here Ho= the variable rr is exogenous is not accepted as the pvalue is greater than .05. hence rr is endogenous

U.K

COMMENT_ – here the p value is greater than .05 hence the null hypothesis is accepted that the rrf is exogenous.

Similarly we can test for rr

Here the p value is lower than .05 , hence we reject the null hypothesis that rr is exogenous.

TEST FOR OVERIDENTIFYING RESTRICTIONS

FRANCE

here we perform the regression of err_1 or the IV residuals on all the instruments.

We calculate the rsquare and multiply it with the sample size to get the test statistic.

Here the test statistic is .869*115=99,935 which is higher than the critical value of chi square at 5% level of significance and 3 d.f i.e 7.814. which means there are misspecification error or the instruments are invalid  here.

Here we perform the regression of err2 or the IV residuals on all the instruments.

Now here the r square is .999*115= 114885 which is much higher than the critical value of chisquare at 5% level and 3 d.f i.e 7.814.  hence the same conclusion as above applies here.

U.K

When rrf is used as an explanatory variable

here we perform the regression of err_1 or the IV residuals on all the instruments.

We calculate the r square and multiply it with the sample size to get the test statistic.

Hence here also the test statistic is greater than the critical value of chi square. therefore there is some misspecification error or the instruments are not valid here.

When rr is used as an explanatory variable

here we perform the regression of err_2 or the IV residuals on all the instruments.

We calculate the r square and multiply it with the sample size to get the test statistic.

France

Using rrf as the explanatory variable

There is a presence of heteroskedasticity since p value of pre32 is less than .05.

similarly using rr as the dependent variable we have

Since the p value of pre42 is less than .05 we reject the null hypothesis, hence there is a presence of heteroskedasticity.

U.K

Using rrf as the explanatory variable

Here the p value of pre_32 is greater than .05. hence the null hypothesis is not rejected and there is no presence of heteroskedacticity.

Using rr as the explanatory variable.

Here the p value of pre_42 is greater than .05. hence we do not reject the null hypothesis. Hence there is no presence of heteroskedasticity.

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