Introduction to Time Series Econometrics - Example

Econometric Report (Name of the student) (Name of the Institution) (Date of Submission) Part1 Q1 The table below shows the return of the stock price the formula below is was used to calculate the returns RT = 100 × log (Rt/Rt‐1). Table 1.0: Daily stock return P Return 7125.58 -0.0071 7818.22 0.0250 7918.27 0.0044 7999.07 -0.0065 7993.16 7649.11 7080.53 -0.0063 7855.62 0.0048 7915.64 -0.0003 8010.05 0.0014 7972.08 7370.64 -0.0364 6983.55 -0.0137 7759.34 -0.0123 7896.09 -0.0025 8035.68 0.0032 8043.25 7387.27 0.0023 7198.96 0.0308 7737.37 -0.0028 7940.2 0.0056 8051.81 0.0020 8040.42 7476.86 0.0121 7553.88 0.0493 7627.96 -0.0141 7940.97 0.0001 8039.72 -0.0015 7999.7 7653.59 0.0236 7574.1 0.0027 7556.53 -0.0094 7874.77 -0.0083 8039.08 -0.0001 7931.96 7704.07 0.0066 7563.89 -0.0013 7482.53 -0.0098 7875.21 0.0001 8009.27 -0.0037 7798.97 7637.23 -0.0087 7722.55 0.0210 7554.73 0.0096 7915.22 0.0051 7927.46 -0.0102 7689.63 7644.99 0.0010 7720.39 -0.0003 7486.08 -0.0091 7873.65 -0.0053 7861.51 -0.0083 7812.33 7717.19 0.0094 7838.59 0.0153 7455.18 -0.0041 7827.31 -0.0059 7889.44 0.0036 7830.47 7769 0.0067 7843.25 0.0006 7455.18 0.0000 7838.31 0.0014 7810.13 -0.0101 7789.98 7769 0.0000 7816.06 -0.0035 7614.04 0.0213 7710.26 -0.0163 7864.76 0.0070 7727.35 7678.57 -0.0116 7773.13 -0.0055 7663.54 0.0065 7765.29 0.0071 7942.33 0.0099 7615.16 7709.61 0.0040 7835.16 0.0080 7749.61 0.0112 7671.43 -0.0121 7941.89 -0.0001 7622.59 7664.33 -0.0059 7751.28 -0.0107 7851.29 0.0131 7676.03 0.0006 7871.99 -0.0088 7513.44 7637.72 -0.0035 7721.99 -0.0038 7822.12 -0.0037 7790.65 0.0149 7873.7 0.0002 7513.44 7440.97 -0.0258 7967.94 0.0319 7792.2 -0.0038 7823.54 0.0042 8018.39 0.0184 7556.53 7462.42 0.0029 7894.69 -0.0092 7851.27 0.0076 7843.19 0.0025 7961.26 -0.0071 7637.51 7499.5 0.0050 7839 -0.0071 7791.21 -0.0076 7908.55 0.0083 8001.99 0.0051 7628.24 7499.73 0.0000 7936.64 0.0125 7722.08 -0.0089 7969.63 0.0077 7982.24 -0.0025 7684.35 7297.99 -0.0269 7941.24 0.0006 7691.51 -0.0040 7996.6 0.0034 8011.12 0.0036 7590.55 7468.06 0.0233 7977.76 0.0046 7800.3 0.0141 7901.76 -0.0119 8075.62 0.0081 7572.99 7492.14 0.0032 7994.1 0.0020 7857.78 0.0074 7881.38 -0.0026 7936.31 -0.0173 7660.5 7500.8 0.0012 7870.11 -0.0155 7887.6 0.0038 7940.65 0.0075 7885.64 -0.0064 7874.58 7539.03 0.0051 7724.22 -0.0185 7768.9 -0.0150 8036.88 0.0121 7846.93 -0.0049 7775.71 7498.29 -0.0054 7318.05 -0.0526 7850.19 0.0105 7985.5 -0.0064 7702.04 -0.0185 7768.15 7470.15 -0.0038 7465.81 0.0202 7788.28 -0.0079 7944.62 -0.0051 7478.75 -0.0290 7874.74 7551.19 0.0108 7641.91 0.0236 7785.78 -0.0003 8006.49 0.0078 7641.22 0.0217 7880.78 7394.39 -0.0208 7624.43 -0.0023 7688.05 -0.0126 7978.91 -0.0034 7703.05 0.0081 7859.96 7356.71 -0.0051 7713.99 0.0117 7670.47 -0.0023 8025.26 0.0058 7566.66 -0.0177 7876.79 7226.63 -0.0177 7713.99 0.0000 7736.95 0.0087 8014.91 -0.0013 7519.68 -0.0062 7785.71 7232.69 0.0008 7769.07 0.0071 7720.28 -0.0022 8034.11 0.0024 7643.54 0.0165 7628.9 7309.2 0.0106 7957.98 0.0243 7710.86 -0.0012 8047.26 0.0016 7671.35 0.0036 7708.4 7210.65 -0.0135 7891.37 -0.0084 7691.71 -0.0025 8062.04 0.0018 7704.56 0.0043 7629.7 7127.58 -0.0115 7835.23 -0.0071 7743.24 0.0067 8105.56 0.0054 7757.13 0.0068 7416.41 6935.05 -0.0270 7699.64 -0.0173 7803.85 0.0078 8074 -0.0039 7684.87 -0.0093 7465.77 7614.63 0.0055 7760.75 0.0079 7803.85 0.0000 8015.22 -0.0073 7629.57 -0.0072 7572.87 7565.77 -0.0064 7701.69 -0.0076 7749.45 -0.0070 8049.61 0.0043 7726.58 0.0127 7572.87 7474.59 -0.0121 7527.79 -0.0226 7738.86 -0.0014 8073.89 0.0030 7891.71 0.0214 7643.92 7554.57 0.0107 7486.18 -0.0055 7857.7 0.0154 8073.89 0.0000 8010.7 0.0151 7715.55 7627.19 0.0096 7572.99 0.0116 7883.51 0.0033 8051.27 -0.0028 7983.22 -0.0034 7775.63 From the two distribution table, the mean variance of a distribution is equavalent to degree of freedom. From the two table, there is no upward distortion in the test and the distibution is evenly throughout the study. Therefore J B model is fit. Q1 (ii) Q1iii Dependent Variable: P Method: ML - ARCH (Marquardt) - Normal distribution Date: 05/24/15 Time: 10:31 Sample (adjusted): 2 260 Included observations: 259 after adjustments Failure to improve Likelihood after 112 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob.   RETURN 3021.612 427.4871 7.068313 0.0000 C 7807.335 5.825302 1340.246 0.0000 Variance Equation C 11086.32 1502.877 7.376731 0.0000 RESID(-1)^2 1.099698 0.289700 3.795994 0.0001 GARCH(-1) -0.502264 0.093870 -5.350657 0.0000 R-squared -0.130562     Mean dependent var 7702.254 Adjusted R-squared -0.134961     S.D. dependent var 265.6284 S.E. of regression 282.9860     Akaike info criterion 13.08023 Sum squared resid 20580831     Schwarz criterion 13.14889 Log likelihood -1688.889     Hannan-Quinn criter. 13.10783 Durbin-Watson stat 0.058523 The equation will be GARCH = C (3) + C (4)*RESID (-1) ^2 + C (5)*GARCH (-1) From the above result indicates that P-value is 0.000 which less that critical value 0.05 hence the null hypotheses is accepted while rejecting alternative hypothesis. Part II Q2 (i) P Rt 7370.64 6951.99 0.018978 6951.99 0.018978 7624.43 -0.00229 7800.3 0.014144 7387.27 0.002256 7060.72 0.01564 7060.72 0.01564 7713.99 0.011746 7857.78 0.007369 7476.86 0.012128 7136.68 0.010758 7136.68 0.010758 7769.07 0.00714 7887.6 0.003795 7653.59 0.023637 7187.44 0.007113 7187.44 0.007113 7957.98 0.024316 7768.9 -0.01505 7704.07 0.006596 7301.67 0.015893 7301.67 0.015893 7891.37 -0.00837 7850.19 0.010464 7637.23 -0.00868 7193.25 -0.01485 7193.25 -0.01485 7835.23 -0.00711 7788.28 -0.00789 7644.99 0.001016 6967.74 -0.03135 6967.74 -0.03135 7699.64 -0.01731 7785.78 -0.00032 7717.19 0.009444 7011.16 0.006232 7011.16 0.006232 7760.75 0.007937 7688.05 -0.01255 7769 0.006714 7176.32 0.023557 7176.32 0.023557 7701.69 -0.00761 7670.47 -0.00229 7678.57 -0.01164 7125.58 -0.00707 7125.58 -0.00707 7527.79 -0.02258 7736.95 0.008667 7709.61 0.004042 7080.53 -0.00632 7080.53 -0.00632 7486.18 -0.00553 7720.28 -0.00215 7664.33 -0.00587 6983.55 -0.0137 6983.55 -0.0137 7572.99 0.011596 7710.86 -0.00122 7637.72 -0.00347 7198.96 0.030845 7198.96 0.030845 7614.63 0.005498 7691.71 -0.00248 7440.97 -0.02576 7553.88 0.049302 7553.88 0.049302 7565.77 -0.00642 7743.24 0.006699 7462.42 0.002883 7574.1 0.002677 7574.1 0.002677 7474.59 -0.01205 7803.85 0.007827 7499.5 0.004969 7563.89 -0.00135 7563.89 -0.00135 7554.57 0.0107 7749.45 -0.00697 7499.73 3.07E-05 7722.55 0.020976 7722.55 0.020976 7627.19 0.009613 7738.86 -0.00137 7297.99 -0.0269 7720.39 -0.00028 7720.39 -0.00028 7818.22 0.025046 7857.7 0.015356 7468.06 0.023304 7838.59 0.01531 7838.59 0.01531 7855.62 0.004784 7883.51 0.003285 7492.14 0.003224 7843.25 0.000594 7843.25 0.000594 7759.34 -0.01226 7918.27 0.004409 7500.8 0.001156 7816.06 -0.00347 7816.06 -0.00347 7737.37 -0.00283 7915.64 -0.00033 7539.03 0.005097 7773.13 -0.00549 7773.13 -0.00549 7627.96 -0.01414 7896.09 -0.00247 7498.29 -0.0054 7835.16 0.00798 7835.16 0.00798 7556.53 -0.00936 7940.2 0.005586 7470.15 -0.00375 7751.28 -0.01071 7751.28 -0.01071 7482.53 -0.00979 7940.97 9.7E-05 7551.19 0.010849 7721.99 -0.00378 7721.99 -0.00378 7554.73 0.009649 7874.77 -0.00834 7394.39 -0.02076 7967.94 0.031851 7967.94 0.031851 7486.08 -0.00909 7875.21 5.59E-05 7356.71 -0.0051 7894.69 -0.00919 7894.69 -0.00919 7455.18 -0.00413 7915.22 0.00508 7226.63 -0.01768 7839 -0.00705 7839 -0.00705 7614.04 0.021309 7873.65 -0.00525 7232.69 0.000839 7936.64 0.012456 7936.64 0.012456 7663.54 0.006501 7827.31 -0.00589 7309.2 0.010578 7941.24 0.00058 7941.24 0.00058 7749.61 0.011231 7838.31 0.001405 7210.65 -0.01348 7977.76 0.004599 7977.76 0.004599 7851.29 0.013121 7710.26 -0.01634 7127.58 -0.01152 7994.1 0.002048 7994.1 0.002048 7822.12 -0.00372 7765.29 0.007137 6935.05 -0.02701 7870.11 -0.01551 7870.11 -0.01551 7792.2 -0.00383 7671.43 -0.01209 6993.08 0.008368 7724.22 -0.01854 7724.22 -0.01854 7851.27 0.007581 7676.03 0.0006 6971.6 -0.00307 7318.05 -0.05258 7318.05 -0.05258 7791.21 -0.00765 7790.65 0.014932 6928.07 -0.00624 7465.81 0.020191 7465.81 0.020191 7722.08 -0.00887 7823.54 0.004222 6822.51 -0.01524 7641.91 0.023588 7641.91 0.023588 7691.51 -0.00396 7843.19 0.002512 Q2 (ii) Return From the two distribution table, the mean variance of a distribution is equavalent to degree of freedom. From the two table, there is no upward distortion in the test and the distibution is evenly throughout the study. Therefore J B model is fit. Q (ii) Q (iii) Included observations: 300 after adjustments Convergence achieved after 18 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob.   RT 2055.875 233.7725 8.794344 0.0000 C 7745.154 9.974774 776.4741 0.0000 Variance Equation C 35531.35 5961.504 5.960132 0.0000 RESID(-1)^2 1.394277 0.188609 7.392432 0.0000 GARCH(-1) -0.826037 0.036022 -22.93168 0.0000 R-squared -0.024163     Mean dependent var 7682.143 Adjusted R-squared -0.027599     S.D. dependent var 265.4799 S.E. of regression 269.1185     Akaike info criterion 13.21235 Sum squared resid 21582578     Schwarz criterion 13.27408 Log likelihood -1976.852     Hannan-Quinn criter. 13.23705 Durbin-Watson stat 0.073302 The equation is as follows GARCH = C (3) + C (4)* RESID (-1) ^2 + C (5)*GARCH (-1) From the above result indicates that P-value is 0.000 which less that critical value 0.05 hence the null hypotheses is accepted while rejecting alternative hypothesis. Q (IV) Included observations: 300 after adjustments Failure to improve Likelihood after 15 iterations Bollerslev-Wooldridge robust standard errors & covariance Presample variance: backcast (parameter = 0.7) GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob.   LOG(GARCH) 1.149654 5.694559 0.201886 0.8400 RT 3063.358 336.3162 9.108564 0.0000 C 7728.239 59.26855 130.3936 0.0000 Variance Equation C 40611.50 2546.201 15.94984 0.0000 RESID(-1)^2 1.169458 0.004290 272.6188 0.0000 GARCH(-1) -0.875012 0.006114 -143.1113 0.0000 R-squared -0.010758     Mean dependent var 7682.143 Adjusted R-squared -0.017565     S.D. dependent var 265.4799 S.E. of regression 267.8013     Akaike info criterion 13.24912 Sum squared resid 21300105     Schwarz criterion 13.32320 Log likelihood -1981.368     Hannan-Quinn criter. 13.27877 Durbin-Watson stat 0.064694 The new equation gives ARCH = C (4) + C (5)*RESID (-1) ^2 + C (6)* ARCH (-1) From the analysis, the p-value gives 0.84 which is greater than critical value which 0.05. Therefore, the null hypothesis is rejected and accepting alternative hypothesis. The graph below explains the conditional standard deviation Q (V) Convergence achieved after 18 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob.   RT 2055.875 233.7725 8.794344 0.0000 C 7745.154 9.974774 776.4741 0.0000 Variance Equation C 35531.35 5961.504 5.960132 0.0000 RESID(-1)^2 1.394277 0.188609 7.392432 0.0000 GARCH(-1) -0.826037 0.036022 -22.93168 0.0000 R-squared -0.024163     Mean dependent var 7682.143 Adjusted R-squared -0.027599     S.D. dependent var 265.4799 S.E. of regression 269.1185     Akaike info criterion 13.21235 Sum squared resid 21582578     Schwarz criterion 13.27408 Log likelihood -1976.852     Hannan-Quinn criter. 13.23705 Durbin-Watson stat 0.073302 The result shows high volatile stock returns indicated with high better of 8.79 Q2 (VI) Included observations: 300 after adjustments Failure to improve Likelihood after 1 iteration Unable to evaluate derivatives at current parameter values Presample variance: backcast (parameter = 0.7) GARCH = 63653.1901467*(1 - C(4) - C(5)) + C(4)*RESID(-1)^2 + C(5)         *GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob.   LOG(GARCH) -50.69664 NA NA NA RT 4637.264 NA NA NA C 8210.985 NA NA NA Variance Equation C -4.05E+09     --     --     -- RESID(-1)^2 63702.14 NA NA NA GARCH(-1) 0.600000 NA NA NA T-DIST. DOF 20.00000 NA NA NA Mean dependent var 7682.143     S.D. dependent var 265.4799 The three conditional variation shows the same result hence the conclusion that the stock are very volatile with high premium prices Read More