The error variance may not be constant through time this is

Question

The capital asset pricing model (CAPM) is an important model in the field of finance. It explains variations in the rate of return on a security as a function of the rate of return on a portfolio consisting of all publicly traded stocks, which is called the market portfolio. Generally the rate of return on any investment is measured relative to its opportunity cost, which is the return on a risk free asset. The resulting difference is called the risk premium since it is the reward or punishment for making a risky investment. The CAPM says that the risk premium on security j is proportional to the risk premium on the market portfolio. That is,

(1) r_j - r_rf = β_j(r_m - r_rf)        

where r_j is the return to security j, r_jf  is the risk free rate, r_m is the return on the market portfolio and β_j is the "beta" of security j. A stock's beta is important to investors since it reveals the stock's volatility. It measures the sensitivity of security j return to variation in the whole stock market. As such, values of beta less than 1 indicate that the stock is "defensive" since its variation is less than the market's. A beta greater than 1 indicates an "aggressive" stock. Investors usually want an estimate of the stock's beta before purchasing it. The CAPM model is the theoretical model. The econometric model is obtained by including an intercept (even though the theoretical model says that it should be zero) and an error term in the model:

(2) r_j - r_rf = α_j + β_j(r_m - r_rf) + ε         

PART 1

1. Explain why the econometric model (equation 2) is a simple regression model like those discussed in class.

2. Choose a public corporation whose stocks are traded in the NYSE. For example, you may choose MICROSOFT, GE, GM, IBM, DISNEY, and MOBIL-EXXON. Using the Finance portal in Yahoo, download monthly data on the stock of your chosen public corporation for the period December 1997 to December 2008 and construct the monthly returns (January 1998 to December 2008) as log differences (not in percent).I attach at the end of this file the return series on the market portfolio (represented by the S&P 500) MKT and the return series on the risk free asset RISKFREE. The 132 observations cover the period January 1998 to December 2008. Estimate the CAPM model for each corporation and comment on their estimated beta values. Interpret your estimates. Does your stock appear aggressive? Defensive?

3. Finance theory (equation 1) says that the intercept parameter should be zero. Does this seem correct given your estimates? Draw a scatter diagram of your data and plot the fitted regression line along with the scatter diagram.

4. Estimate the model (equation 2) under the assumption that the intercept is zero. Do the estimates of beta change much?

5. What is the standard error of β_j? Test the null that β_j is zero (at alpha = 0.01). What is RSS, ESS, TSS, R-square? Construct the ANOVA test.

PART 2-

There are, however, potential problems with the previous econometric findings.

6. The beta may not be constant through time. Use the dummy variable approach and the Chow test approach to determine whether the assumption of parameter constancy is valid. Use 2001m10 as the break date.

7. The error variance may not be constant through time. This is known as heteroskedasticity. Use the White test to investigate the issue of heteroskedastic disturbances.

8. The errors may be correlated through time. This is known as autocorrelation or serial correlation. Use the Durbin Watson test to test the hypothesis of first order positive autocorrelation.

9. Returns may be non-linearly related to market returns rather than the linear relation that is suggested in the statistical model. Construct a test for linearity.

10. To add some dynamic content, re-estimate the model by including the one month lag of the market risk premium. Is the lagged market risk premium significant?

date	MKT	RISKFREE
19980130	0.004529	0.004188
19980227	0.07323	0.004268
19980331	0.051322	0.004358
19980430	0.010862	0.00394
19980529	-0.025755	0.003806
19980630	0.031954	0.003919
19980731	-0.023264	0.003954
19980831	-0.157667	0.003909
19980930	0.063836	0.003357
19981030	0.074357	0.00296
19981130	0.061986	0.003702
19981231	0.063053	0.003662
19990129	0.038346	0.003674
19990226	-0.038105	0.003755
19990331	0.037862	0.003798
19990430	0.049	0.00371
19990528	-0.02072	0.003683
19990630	0.05102	0.003698
19990730	-0.030635	0.003708
19990831	-0.009983	0.00375
19990930	-0.022867	0.003751
19991029	0.062048	0.003638
19991130	0.036856	0.003809
19991231	0.083925	0.004252
20000131	-0.039753	0.004488
20000229	0.03177	0.004428
20000331	0.053552	0.00488
20000428	-0.059467	0.004392
20000531	-0.039047	0.003896
20000630	0.051649	0.004691
20000731	-0.017105	0.004896
20000831	0.075811	0.005195
20000929	-0.051139	0.004933
20001031	-0.024564	0.005023
20001130	-0.102546	0.005099
20001229	0.020346	0.004808
20010131	0.039533	0.004023
20010228	-0.099267	0.004178
20010330	-0.07029	0.003693
20010430	0.083904	0.003118
20010531	0.010561	0.002871
20010629	-0.017481	0.002846
20010731	-0.018321	0.003033
20010831	-0.05908	0.002812
20010928	-0.09154	0.001982
20011031	0.027967	0.001744
20011130	0.078734	0.00144
20011231	0.017841	0.001363
20020131	-0.01606	0.001408
20020228	-0.021705	0.001444
20020328	0.044693	0.001425
20020430	-0.049649	0.001461
20020531	-0.010458	0.001408
20020628	-0.070243	0.001389
20020731	-0.081137	0.001417
20020830	0.007984	0.00138
20020930	-0.099975	0.001308
20021031	0.074958	0.001183
20021129	0.061276	0.001014
20021231	-0.053309	0.000947
20030131	-0.02343	0.000955
20030228	-0.015408	0.000996
20030331	0.010333	0.000951
20030430	0.082797	0.000911
20030530	0.063507	0.000934
20030630	0.016332	0.000662
20030731	0.02313	0.000733
20030829	0.024908	0.000809
20030930	-0.009102	0.00071
20031031	0.060331	0.000763
20031128	0.016607	0.00075
20031231	0.045532	0.000689
20040130	0.023064	0.000689
20040227	0.015467	0.000778
20040331	-0.010685	0.000769
20040430	-0.02423	0.000674
20040528	0.014127	0.000757
20040630	0.021563	0.000941
20040730	-0.037681	0.001065
20040831	0.002714	0.001201
20040930	0.020555	0.001215
20041029	0.01781	0.001457
20041130	0.048243	0.001693
20041231	0.03518	0.00159
20050131	-0.026557	0.001802
20050228	0.022695	0.002081
20050331	-0.016937	0.002166
20050429	-0.025182	0.002259
20050531	0.037916	0.002293
20050630	0.011531	0.002488
20050729	0.043331	0.002691
20050831	-0.005943	0.002784
20050930	0.010609	0.002655
20051031	-0.020784	0.00308
20051130	0.040406	0.003309
20051230	0.003476	0.003288
20060131	0.040112	0.003606
20060228	-0.001644	0.003687
20060331	0.01906	0.003828
20060428	0.012968	0.003844
20060531	-0.031042	0.003913
20060630	-0.000396	0.003903
20060731	-0.001902	0.0042
20060831	0.025085	0.004138
20060929	0.019452	0.003874
20061031	0.037086	0.004293
20061130	0.023716	0.00415
20061229	0.010823	0.003973
20070131	0.019438	0.004168
20070228	-0.013998	0.004286
20070330	0.012949	0.004209
20070430	0.039893	0.003963
20070531	0.038894	0.003913
20070629	-0.014758	0.003776
20070731	-0.031753	0.004254
20070831	0.011623	0.00346
20070928	0.040903	0.002869
20071031	0.025842	0.003318
20071130	-0.049279	0.002683
20071231	-0.004342	0.002382
20080131	-0.062305	0.001387
20080229	-0.02204	0.001725
20080331	-0.01047	0.001032
20080430	0.051144	0.001048
20080530	0.023825	0.001623
20080630	-0.078625	0.001412
20080731	-0.01315	0.001319
20080829	0.011042	0.001387
20080930	-0.09806	0.000803
20081031	-0.184726	0.000212
20081128	-0.085206	0.000025
20081231	0.021482	0.000025