1. Economics 310 Lecture 16 Autocorrelation Continued

2. Bid-Ask Spread an Example The spread between the bid price for US currency and the ask price for US currency in the Brazilian blackmarket is function of opportunity cost of holding currency and the risk of holding currency. Opportunity cost is interest rate risk is the rate of variability in exchange rate

3. Breusch-Godfrey Test Test for higher order autocorrelation Estimate model by ols regress residual on independent variables, lagged values of residual n times r-square is chi-square

4. Breusch-Godfrey Test

5. Breusch-Godfrey Example |_?ols spread interest sigma / resid=e |_sample 13 58 |_genr e1=lag(e) |_genr e2=lag(e1) |_genr e3=lag(e2) |_genr e4=lag(e3) |_genr e5=lag(e4) |_genr e6=lag(e5) |_genr e7=lag(e6) |_genr e8=lag(e7) |_genr e9=lag(e8) |_genr e10=lag(e9) |_genr e11=lag(e10) |_genr e12=lag(e11) |_ols e interest sigma e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12

6. Breusch-Godfrey Results VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 31 DF P-VALUE CORR. COEFFICIENT AT MEANS INTEREST 0.23151E-01 0.7471E-01 0.3099 0.759 0.056 0.0768 -8.2815 SIGMA -0.25661E-01 0.1656 -0.1549 0.878-0.028 -0.0417 3.6503 E1 0.29622 0.1780 1.664 0.106 0.286 0.3041 -0.3859 E2 -0.11449 0.1904 -0.6013 0.552-0.107 -0.1168 0.2242 E3 0.10998 0.1830 0.6009 0.552 0.107 0.1119 -0.2605 E4 0.28617 0.1863 1.536 0.135 0.266 0.2870 -0.9687 E5 -0.27742 0.1981 -1.400 0.171-0.244 -0.2778 1.0122 E6 0.37118 0.2702 1.374 0.179 0.240 0.2898 0.1029 E7 -0.36543 0.2575 -1.419 0.166-0.247 -0.2840 -0.2710 E8 -0.30811 0.2663 -1.157 0.256-0.203 -0.2359 0.0288 E9 0.14887 0.2859 0.5207 0.606 0.093 0.1139 -0.0193 E10 0.25463 0.3172 0.8028 0.428 0.143 0.1757 0.4847 E11 -0.35927E-01 0.2957 -0.1215 0.904-0.022 -0.0247 -0.0543 E12 -0.20655 0.2981 -0.6930 0.493-0.124 -0.1417 -0.2422 CONSTANT -0.10996 0.3376 -0.3257 0.747-0.058 0.0000 5.9803 |_gen1 lm=$n*$r2..NOTE..CURRENT VALUE OF $N = 46.000..NOTE..CURRENT VALUE OF $R2 = 0.27190 |_print lm LM 12.50750

7. Obtaining Efficient Estimates When rho known, we can use generalized least squares. When rho unknown: First difference method No intercept Berenblutt-Webb test rho based on Durbin Watson statistic Cochrane-Orcutt Iterative Two step Durbin two-step

8. When structure of autocorrelation is known

9. Rho unknown - 1st difference

10. Estimating Rho

11. Estimating Rho-continued

12. Estimating our model with Cochrane-Orcutt |_auto spread interest sigma LEAST SQUARES ESTIMATION 58 OBSERVATIONS BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = 0.00100 ITERATION RHO LOG L.F. SSE 1 0.00000 -64.8077 31.732 2 0.23843 -63.1166 29.904 3 0.24368 -63.1169 29.903 4 0.24380 -63.1169 29.903 ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC ESTIMATE VARIANCE ST.ERROR T-RATIO RHO 0.24380 0.01622 0.12734 1.91448 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 55 DF P-VALUE CORR. COEFFICIENT AT MEANS INTEREST 0.20675 0.5310E-01 3.894 0.000 0.465 0.4739 0.3455 SIGMA 0.36489 0.1014 3.598 0.001 0.436 0.3831 0.2599 CONSTANT 1.3748 0.3088 4.452 0.000 0.515 0.0000 0.3933

13. Estimating rho-Cochrane- Orcutt two step |_ols spread interest sigma / resid=e dw |_sample 2 58 |_genr e1=lag(e) |_ols e e1 / noconst VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 56 DF P-VALUE CORR. COEFFICIENT AT MEANS E1 0.23843 0.1304 1.828 0.073 0.237 0.2373 0.5812

14. Estimating rho Durbin-Watson Statistic |_ols spread interest sigma / resid=e dw REQUIRED MEMORY IS PAR= 32 CURRENT PAR= 500 OLS ESTIMATION 58 OBSERVATIONS DEPENDENT VARIABLE = SPREAD...NOTE..SAMPLE RANGE SET TO: 1, 58 DURBIN-WATSON STATISTIC = 1.51549

15. Estimating rho Durbin Two-step |_genr y1=lag(spread) |_genr x1=lag(interest) |_genr x2=lag(sigma) |_ols spread interest x1 sigma x2 y1 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 51 DF P-VALUE CORR. COEFFICIENT AT MEANS INTEREST 0.23253 0.1103 2.109 0.040 0.283 0.5364 0.3881 X1 -0.74972E-01 0.1095 -0.6844 0.497-0.095 -0.1698 -0.1229 SIGMA 0.36044 0.1085 3.322 0.002 0.422 0.3758 0.2582 X2 -0.10560 0.1217 -0.8678 0.390-0.121 -0.1105 -0.0739 Y1 0.24587 0.1382 1.779 0.081 0.242 0.2472 0.2433 CONSTANT 1.0809 0.3163 3.418 0.001 0.432 0.0000 0.3073

16. Comparing estimates of rho