The ARCH Model: Empirical Analysis - Assignment Example

The ARCH Model: Empirical Analysis 1.1. Unit Root Test Unit root tests was used in this project to establish whether the time series parameter (Year) is not stationary or not. It used three fundamental functions such as Dickey Fuller function, autocorrelation function and Phillips Perron. 1.1.1. Autocorrelation corrgram STY, lags(20) -1 0 1 -1 0 1 LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] ------------------------------------------------------------------------------- 1 0.6210 0.6232 15.84 0.0001 |---- |---- 2 0.1407 -0.3701 16.676 0.0002 |- --| 3 -0.1985 -0.1601 18.387 0.0004 -| -| 4 -0.2829 -0.0386 21.965 0.0002 --| | 5 -0.1797 0.1318 23.453 0.0003 -| |- 6 -0.0176 -0.0522 23.468 0.0007 | | 7 0.0747 -0.0260 23.741 0.0013 | | 8 0.0763 0.0014 24.037 0.0023 | | 9 0.0855 0.2243 24.42 0.0037 | |- 10 0.0712 -0.0261 24.695 0.0060 | | 11 -0.0059 -0.0550 24.697 0.0101 | | 12 -0.1257 -0.1403 25.621 0.0121 -| -| 13 -0.2079 -0.1036 28.25 0.0083 -| | 14 -0.1589 0.1621 29.849 0.0080 -| |- 15 -0.0390 0.1872 29.949 0.0121 | |- 16 0.0881 0.1674 30.485 0.0156 | |- 17 0.1969 0.1819 33.292 0.0103 |- |- 18 0.1880 . 35.979 0.0071 |- 19 0.0213 . 36.015 0.0105 | 20 -0.1604 . 38.189 0.0084 -| . corrgram ITY, lags(20) -1 0 1 -1 0 1 LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] ------------------------------------------------------------------------------- 1 0.6403 0.6471 16.845 0.0000 |----- |----- 2 0.2486 -0.2797 19.454 0.0001 |- --| 3 -0.0468 -0.1228 19.549 0.0002 | | 4 -0.2373 -0.1316 22.065 0.0002 -| -| 5 -0.2781 0.0071 25.626 0.0001 --| | 6 -0.3609 -0.3071 31.813 0.0000 --| --| 7 -0.3629 -0.0690 38.271 0.0000 --| | 8 -0.2042 0.0384 40.383 0.0000 -| | 9 0.0101 0.1138 40.388 0.0000 | | 10 0.1686 0.0808 41.932 0.0000 |- | 11 0.1209 -0.3980 42.755 0.0000 | ---| 12 -0.0057 -0.1677 42.757 0.0000 | -| 13 -0.1624 -0.4008 44.361 0.0000 -| ---| 14 -0.2582 -0.1312 48.581 0.0000 --| -| 15 -0.1900 -0.2981 50.968 0.0000 -| --| 16 -0.1273 -0.3561 52.087 0.0000 -| --| 17 0.0219 0.3508 52.122 0.0000 | |-- 18 0.2168 . 55.696 0.0000 |- 19 0.2583 . 61.035 0.0000 |-- 20 0.2808 . 67.691 0.0000 |-- The autocorrelation test shows 4 fields including the lag, autocorrelation (AC), partial autocorrelation (PAC), Q-statistic (Q) and p-value (P). The function shows that the p-values for the lags are significant at 1% level. 1.1.2. Dickey-Fuller . dfuller STY, lags(0) Dickey-Fuller test for unit root Number of obs = 37 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -2.930 -3.668 -2.966 -2.616 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0420 . dfuller ITY, lags(0) Dickey-Fuller test for unit root Number of obs = 37 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -2.716 -3.668 -2.966 -2.616 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0714 . dfuller STY, lags(0) Dickey-Fuller test for unit root Number of obs = 37 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -2.930 -3.668 -2.966 -2.616 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0420 . dfuller ITY, lags(0) Dickey-Fuller test for unit root Number of obs = 37 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(t) -2.716 -3.668 -2.966 -2.616 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.0714 The Dickey Fuller function tests the null hypothesis of any unit root. The null hypothesis of this study’s model is that there is no relationship between two variables STY and ITY and that ITY carries a non-stationery unit root. The rejection of the null hypothesis happens if the absolute test has statistical values less than the critical value of Dickey Fuller. The p - value of Z (t) is significant at 5% level, causing the null hypothesis to be rejected. The time-series variable is therefore non-stationary. 1.1.3. Phillip-Perron pperron STY, lags(4) trend regress Phillips-Perron test for unit root Number of obs = 37 Newey-West lags = 4 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(rho) -11.782 -24.036 -18.812 -16.176 Z(t) -2.898 -4.270 -3.552 -3.211 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.1630 ------------------------------------------------------------------------------ STY | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- STY | L1. | .5925958 .1304633 4.54 0.000 .3274625 .857729 _trend | -.0286995 .0242134 -1.19 0.244 -.0779072 .0205081 _cons | 7.542043 2.375563 3.17 0.003 2.714318 12.36977 ------------------------------------------------------------------------------ . . pperron ITY, lags(4) trend regress Phillips-Perron test for unit root Number of obs = 37 Newey-West lags = 4 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(rho) -13.446 -24.036 -18.812 -16.176 Z(t) -2.704 -4.270 -3.552 -3.211 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.2344 ------------------------------------------------------------------------------ ITY | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ITY | L1. | .6480799 .1320359 4.91 0.000 .3797507 .916409 _trend | .0036617 .0271941 0.13 0.894 -.0516034 .0589269 _cons | 7.723568 3.0022 2.57 0.015 1.622364 13.82477 ------------------------------------------------------------------------------ The Phillips-Perron (PP) test was used in estimation of the non-augmented DF test model. It modifies the t - ratio of alpha coefficient (α) in the equation. The resultant serial correlation does not change the asymptotic distribution in the test statistic. PP test is essential, as it is not necessary to select the serial correlation level. Additionally, it does not need investigators to specify lengths of lags in the tests. It worked well in this test because of the large size of the data. The PP test shows that the p-value for is significant at the 5% confidence level. The null hypothesis is rejected and the time series variable becomes non-stationary. 1.2. Volatility Test 1.2.1. ARCH arch Year ITY STY, arch(1/1) ARCH family regression Sample: 1 - 2017, but with a gap Number of obs = 39 Distribution: Gaussian Wald chi2(2) = 50.96 Log likelihood = -146.0082 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | OPG Year | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Year | ITY | 1.311635 .1841733 7.12 0.000 .9506624 1.672608 STY | .4222038 .1752011 2.41 0.016 .0788159 .7655918 _cons | 1964.597 6.159556 318.95 0.000 1952.525 1976.67 -------------+---------------------------------------------------------------- ARCH | arch | L1. | 2.854949 .4162435 6.86 0.000 2.039127 3.670771 | _cons | .0322902 .2333326 0.14 0.890 -.4250333 .4896138 . arima Year ITY STY Number of gaps in sample: 1 (note: filtering over missing observations) (setting optimization to BHHH) Iteration 0: log likelihood = -256.89584 Iteration 1: log likelihood = -256.89584 The ARCH Model is of great significance as it estimated the weights to be used as parameters and the data to determine the optimal weights. The model allowed the weights to be used for predicting the variances. The basic ARCH (1) model was also vital for the description of the conditional variance of the shock when the time t is generated by functions of the squares of previous shocks. The model is easy to use as it takes care of all the non-linear properties and clustered errors. The result in the ARCH (1) analysis shows the lag coefficient of lag 1 as being statistically significant at the level of confidence of 10%. At the same time, the constant term is seen to be statistically significant at the 1% confidence level. The overall result shows that for each of the variables, the analysis gives a 95% confidence interval. On the ground of the ARCH (1) results, the variance in forecasting error was computed. This generated the line graph in the figure 1 below. There was very low in 1980 and increased to the maximum in 1985 before falling to the lowest value in 1982.From the observation, this indicates existence of volatility clustering in time series variable (Year). The limitation of the ARCH model included the fact that it needed estimation of the p autoregressive terms. This of course uses many degrees of freedom. Since some coefficients are negative, it is difficult to explain all of the coefficients. It is recommended that higher ARCH functions be used, such as ARCH (3) to be estimated by the GARCH model. 1.2.2. GARCH The result above shows that the p-values of ARCH lag 1 and GARCH lag1 are not significant at all. The forecasting error variance generated the above diagram, showing the graph of forecasted error variance. There appears a big shift between 1980 and 1990 and smaller spikes between 2005 and 2013. However, the difference between the forecasted variance of error between ARCH and GARCH was very little. 1.2. Cointegration and error correction The co integration is the regression of a non-stationary variable of time series against a non-stationary variable. The phenomenon however shows that there is no true link between these variables as each of them grows independently. If both are non-stationary, regression on one of them against the other can eliminate the non-stationary time series variable. This can suggest that the existence of a long-run correlation between the two. Test on the co integration requires Engle-Granger EG test. There first step is to confirm that there the time series is not stationary. Secondly, we estimate the ECM term and the co integration equation. The third step is to perform the CRADF test. The fourth step is to conduct the Granger Causality (GC) test. Finally, we estimate the short run dynamic equations using lagged term of one period ECM. . Phillips-Perron test pperron STY, trend Phillips-Perron test for unit root Number of obs = 37 Newey-West lags = 3 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(rho) -14.058 -24.036 -18.812 -16.176 Z(t) -3.052 -4.270 -3.552 -3.211 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.1182 The p-value is significant at 5% confidence level. The absolute test statistics are less than the absolute 1 % critical value. In this regard, the time series parameter is non-stationary. . pperron ITY, trend Phillips-Perron test for unit root Number of obs = 37 Newey-West lags = 3 ---------- Interpolated Dickey-Fuller --------- Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value ------------------------------------------------------------------------------ Z(rho) -14.787 -24.036 -18.812 -16.176 Z(t) -2.824 -4.270 -3.552 -3.211 ------------------------------------------------------------------------------ MacKinnon approximate p-value for Z(t) = 0.1884MacKinnon approximate p-value for Z(t) = 0.1884 The absolute test has smaller statistics than the absolute critical value at the 1% level. The variable is therefore non-stationary and are integrated in the same order. cointegration equation The results of cointegraion equation between ITY and STY are shown below. Source SS df MS Number of obs = 44 ------------------------------------------------------------------------------------- F( 1, 42) = 3.41 Model 287.764041 1 287.764041 Prob> F = 0.0719 Residual 3545.69176 42 84.4212324 R-squared = 0.0751 ------------------------------------------------------------------------------------- Adj R-squared = 0.0530 Total 3833.4558 43 89.1501349 Root MSE = 9.1881 ----------------------------------------------------------------------------------------------------------------------------- ITY Coef. Std. Err. t P>|t| [95% Conf. Interval] ----------------------------------------------------------------------------------------------------------------------------- STY .2421125 .1311368 1.85 0.072 -.0225324 .5067573 _cons 19.43614 6.632752 2.93 0.005 6.050702 32.82157 ----------------------------------------------------------------------------------------------------------------------------- The results show the p-values of STY as well as the constant term to be significant at the 10% and 1% levels of confidence respectively. This is in the 95% confident interval. Co integration equation ITY = 21.39 + 0.272STY + e - CRADF test The co integration equation has been generated and will be used to obtain the ECM (Error Correction Model) term to do the CRADF test. The ECM is appropriate because we have time series data and needs to obtain the short run as well as the long run connection between many time series variables. The ECM term is written as. ECM = ITY – (21.39 + 0.272STY) Dickey-Fuller test for unit root Number of obs = 43 --------------- Interpolated Dickey-Fuller ------------------ Test 1% Critical 5% Critical 10% Critical Statistic Value ValueValue ------------------------------------------------------------------------------------------------- Z(t) -2.911 -2.631 -1.950 -1.607 ------------------------------------------------------------------------------------------------- Long Run Source SS df MS Number of obs = 39 -------------------------------------------------------------------------- F( 2, 40) = 202.67 Model 2136.44025 2 1068.22012 Prob> F = 0.0000 Residual 210.824904 40 5.2706226 R-squared = 0.9102 -------------------------------------------------------------------------- Adj R-squared = 0.9057 Total 2347.26515 42 55.8872655 Root MSE = 2.4958 ------------------------------------------------------------------------------------------------------------ diff_ITYCoef. Std. Err. t P>|t| [95% Conf. Interval] ------------------------------------------------------------------------------------------------------------ ITY .7261984 .0866532 17.71 0.000 .7319088 .920488 ecm1 -.6904142 .0785027 -18.36 0.000 -.9884418 -.7923866 _cons -24.90015 1.707609 -17.18 0.000 -28.94714 -22.85315 ------------------------------------------------------------------------------------------------------------- In the results above, the ECM value is negative, hence it is statistically significant. Source SS df MS Number of obs = 39 ------------------------------------------------------------------------- F( 2, 40) = 6.63 Model 584.602735 2 292.301367 Prob> F = 0.0033 Residual 1762.66242 40 44.0665604 R-squared = 0.2491 ------------------------------------------------------------------------- Adj R-squared = 0.2115 Total 2347.26515 42 55.8872655 Root MSE = 6.6383 ----------------------------------------------------------------------------------------------------------------- diff_ITYCoef. Std. Err. t P>|t| [95% Conf. Interval] ------------------------------------------------------------------------------------------------------------------ STY .1450484 .0957415 1.51 0.138 -.0484524 .3385492 ecm1 -.3492332 .1122627 -3.11 0.003 -.5761245 -.1223418 _cons -8.125461 4.85537 -1.47 0.150 -16.93853 2.687609 ------------------------------------------------------------------------------------------------------------------- In the results above, the ECM value is negative, hence it is statistically significant. - short run dynamic equation The ECM term connects the long run equilibrium link from co integration with the short run dynamic model, whose results are shown below. Source SS df MS Number of obs = 39 ------------------------------------------------------------------------- F( 5, 34) = 2.90 Model 550.078582 5 110.015716 Prob> F = 0.0275 Residual 1289.08661 34 37.914312 R-squared = 0.2991 ------------------------------------------------------------------------- Adj R-squared = 0.1960 Total 1839.16519 39 47.1580818 Root MSE = 6.1575 --------------------------------------------------------------------------------------------------------------------- dITYCoef. Std. Err. t P>|t| [95% Conf. Interval] ---------------------------------------------------------------------------------------------------------------------- ecm1 -1.282091 .4351438 -2.2 0.010 -2.066409 -.2977721 dITY1 .924821 .418391 1.91 0.064 -.04854 1.633504 dITY2 .2204802 .1674919 1.32 0.197 -.1199042 .5608646 dITY4 -.1285627 .1193208 -1.08 0.289 -.3710518 .1139264 dSTY3 -.13577 .1126857 -1.49 0.147 -.3963626 .0616472 _cons -19.739 12.3634 -1.60 0.120 -44.85284 5.398069 ----------------------------------------------------------------------------------------------------------------------- Conclusion 2.1. Summary of the Findings From the findings of these tests, the linear correlation coefficient shows the coefficient of association as 0.7504. The regression analysis from the plotted graph shows a straight line, an indicator of positive relationship between the two variables. The sign rank test gave the mean of the random distribution of ITY as 3.571506 and the standard deviation as 0.0722481. The results of descriptive statistics showed the mean to be 3.79549 and standard deviation as 0.0740257. From the unit root test, the Dickey Fuller Test demonstrated the 1% critical value as -3.668, the 5% critical value as -2.966 and the 10% critical value as -2.616. For the 39 observations the mean obtained as 3.571506 while. The standard deviation ranged between 0.0622481 and 0.0730257 depending on the critical value. The p - value for Z (t) was generated as 0.0714. The Phillips-Perron test on the other hand had the p – value as 0.163. The T - Test on the two variables was to test how close STY was to ITY. It presented a combined mean of 3.72903, a standard error of 0.0070082 and the standard deviation as 0.0943768 for the 39 observations. For the volatility test, the ARCH family regression generated 56 iterations. The Log likelihood in the arch test is -146.0082. The T - Test on the log return of the standard and poor 500 value was conducted, presenting the combined mean of 7.14, standard error of 0.0031645 and standard deviation of 0.0502348 for the 252 observations. 2.2. Reflection The correlation between level of investment and the level of saving is represented by the model below: In this model, we consider αi as the standard deviation 0.0502348 and βi is the mean (7.14) while εi is the standard error (0.0031645). Feldstein and Horioka present the correlation between the two as FH. In the analysis of FH, the existence of high correlation between savings and the investment shows that there is low capital mobility. Taking one instance from the records, the seventh observation in 1986 shows ITY as 24.104 and STY as 19.255. 24.104 = αi + βi * 19.255 + εit FH = STY / ITY FH = 19.255 / 24.104 = 0.79883 Considering the linear correlation coefficient obtained in this test as 0.7504, the error is 0.79883 – 0.7504 = 0.04843, which is less than 0.5. The Feldstein-Horioka puzzle is therefore valid. This overrules the null hypothesis that under the complete mobility of capital there is a zero retention coefficient of savings. Read More