Essays on Economics: ARMA and GARCH Models Term Paper

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 Economics –Time Series 1.Introduction In this essay, the aim is to examine an economic time series and to examine its stationarity properties. Moreover, ARMA and GARCH models are fit and its in sample and forecasting properties are examined. The Vector Auto Regression Models are also fit with two economic series here. 2. The Data The sample used here is annual data for inflation for USA from 1950 to 2009.The inflation is calculated as the rate of change of GDP deflator. There are 60 observation sin the sample. 3. The Stationarity The graph 1 in appendix 2 shows the trend of the inflation. It shows an increasing trend fluctuating around its trend value. The next step is to examine the stationarity property of the series based on the correlogram of the series inflation.

The correlogram given in appendix 2 shows that the series is non stationary. This is because the first PACF reaches 0.8 and then it converges to zero. The stationarity property is examined by first differencing the series. The correlogram of the new series dinflation is also seen from appendix 2.It shows that the ACF oscillates around zero and the PACF is subject to an oscillating decay.

This means the differenced series is stationary. All the autocorrelations are significant for all lags other than the first one. Hence there is autocorrelation problem. 4. ARMA Models Three ARMA models are chosen ARMA (1, 1), MA (1) and AR (1). The results are given in the ARMA model section of appendix 2. In all the cases, the models are significant as shown by the Wald Chi square Statistic. Both the information criteria Akaike Information Criteria (AIC) and the Schwartz Bayesian Criterion (SBC) show that ARMA (1,0,1) is the model with the best fit since both of these have least values in this case. The results show that in the case of ARMA (1, 1) all the coefficients are significant.

In the case of AR (1) and MA (1) however the time variable is not significant. The Durbin Watson (DW) Statistic shows the presence of no autocorrelation in all the cases. The model specification tests show no sign of autocorrelation, heteroskedasticity and omitted variable bias in all the cases. 5. GARCH Models The GARCH models are shown in the ARCH model section of appendix 2.The two GARCH models selected are ARCH(1,1) and ARCH(2,2). In the first case, both ARCH and GARCH terms are significant.

However, the overall model is not significant as shown by the Wald Chi Square Statistic. In the second case also, both the ARCH and GARCH terms are significant. The overall model is significant as shown by the Wald Chi square statistic. Both the values of AIC and SBC show this model as the one with most predictive power. 6. VAR Models The Vector Auto Regression (VAR) models are fitted with the data for real GDP in US and the inflation.

The results are given in the last section of appendix 2. The R squared value gives an overall fit in the case of both inflation and GDP as the dependent variables. In the case of the VAR for inflation, only the first equation is significant. In the case of VAR for real GDP also, the first equation is significant. Appendix 1 Data DATE GDPdeflator Real GDP Inflation 01/01/1950 14.635 2006.83 01/01/1951 15.697 2161.6 7.256577 01/01/1952 15.967 2244.0 1.720074 01/01/1953 16.161 2347.6 1.215006 01/01/1954 16.307 2332.7 0.903409 01/01/1955 16.585 2500.5 1.704789 01/01/1956 17.155 2550.3 3.436841 01/01/1957 17.726 2601.3 3.328476 01/01/1958 18.122 2578.1 2.234007 01/01/1959 18.339 2762.4 1.19744 01/01/1960 18.596 2831.3 1.401385 01/01/1961 18.804 2897.3 1.11852 01/01/1962 19.062 3072.6 1.372048 01/01/1963 19.264 3207.0 1.0597 01/01/1964 19.562 3392.8 1.546927 01/01/1965 19.917 3610.5 1.814743 01/01/1966 20.483 3845.6 2.841794 01/01/1967 21.113 3943.1 3.075721 01/01/1968 22.009 4134.2 4.243831 01/01/1969 23.098 4262.3 4.947976 01/01/1970 24.317 4270.3 5.277513 01/01/1971 25.531 4413.8 4.992392 01/01/1972 26.629 4648.7 4.300654 01/01/1973 28.109 4917.6 5.55785 01/01/1974 30.669 4889.3 9.107404 01/01/1975 33.553 4880.9 9.403632 01/01/1976 35.485 5141.9 5.758054 01/01/1977 37.742 5378.9 6.360434 01/01/1978 40.382 5680.3 6.99486 01/01/1979 43.756 5855.7 8.355207 01/01/1980 47.752 5838.9 9.132462 01/01/1981 52.227 5987.1 9.371335 01/01/1982 55.412 5870.9 6.098378 01/01/1983 57.59 6137.5 3.930557 01/01/1984 59.76 6577.8 3.768015 01/01/1985 61.571 6849.8 3.030455 01/01/1986 62.934 7086.9 2.213705 01/01/1987 64.757 7314.1 2.896685 01/01/1988 66.981 7614.7 3.434378 01/01/1989 69.514 7886.3 3.781669 01/01/1990 72.202 8033.7 3.866847 01/01/1991 74.757 8015.4 3.538683 01/01/1992 76.527 8287.7 2.367671 01/01/1993 78.219 8523.9 2.210984 01/01/1994 79.867 8871.2 2.106905 01/01/1995 81.533 9094.0 2.085968 01/01/1996 83.083 9434.5 1.901071 01/01/1997 84.551 9854.9 1.766908 01/01/1998 85.506 10284.1 1.129496 01/01/1999 86.764 10780.4 1.471242 01/01/2000 88.642 11226.6 2.164492 01/01/2001 90.649 11347.3 2.264164 01/01/2002 92.116 11553.2 1.61833 01/01/2003 94.094 11841.6 2.147293 01/01/2004 96.764 12264.7 2.837588 01/01/2005 99.994 12639.2 3.338018 01/01/2006 103.255 12976.5 3.261196 01/01/2007 106.292 13229.4 2.941262 01/01/2008 108.625 13228.2 2.194897 01/01/2009 109.614 12880.7 0.910472 Appendix 2 1.Graphs of the series Inflation 2.Correlograms 3.

ARMA MODELS ARMA(1,1) Time-series regression -- ARMA disturbances Sample: 1 to 59 Number of obs = 59 Wald chi2(3) = 223.88 Log likelihood = -92.56299 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | OPG inflation | Coef.

Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- inflation | time | -. 1516512 . 0705168 -2.15 0.032 -. 2898616 -. 0134408 _cons | 9.415853 . 5942996 15.84 0.000 8.251047 10.58066 -------------+---------------------------------------------------------------- ARMA | ar | L1.

| . 8853609 . 0688119 12.87 0.000 . 7504921 1.02023 ma | L1.

| . 4639078 . 1224266 3.79 0.000 . 2239561 . 7038595 -------------+---------------------------------------------------------------- SIGMA2 | _cons | 1.349666 . 1881881 7.17 0.000 . 9808242 1.718508 AIC= -8.54 ; SBC=-3.25; DW=1.92 Model Specification Tests Tests Statistic p value (1) (2) (1) (2) Auto Regression (F) 0.4 1.3 0.6 0.3 ARCH (F) 0.2 0.1 0.6 0.9 Normality (2 ) 0.4 2.5 0.8 0.3 Reset (F) 0.5 3.4 0.5 0.1 MA(1) Time-series regression -- MA disturbances Sample: 1 to 59 Number of obs = 59 Wald chi2(2) = 47.40 Log likelihood = -111.5407 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | OPG inflation | Coef.

Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- inflation | time | -. 0307587 . 0298521 -1.03 0.303 -. 0892677 . 0277503 _cons | 4.56434 . 7018667 6.50 0.000 3.188706 5.939973 -------------+---------------------------------------------------------------- ARMA | ma | L1.

| . 719098 . 1047208 6.87 0.000 . 5138491 . 9243469 -------------+---------------------------------------------------------------- SIGMA2 | _cons | 2.568222 . 4819412 5.33 0.000 1.623635 3.51281 AIC=2.43;SBC=3.21;DW=1.75 Tests Statistic p value Auto Regression (F) 1.99 0.15 ARCH (F) 0.36 0.52 Normality (2 ) 0.51 0.78 Reset (F) 1.12 0.30 AR(1) Time-series regression -- AR disturbances Sample: 1 to 59 Number of obs = 59 Wald chi2(2) = 186.08 Log likelihood = -96.25324 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | OPG inflation | Coef.

Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- inflation | time | -. 0910038 . 0957615 -0.95 0.342 -. 2786929 . 0966854 _cons | 6.818494 2.53642 2.69 0.007 1.847203 11.78978 -------------+---------------------------------------------------------------- ARMA | ar | L1.

| . 890403 . 0732921 12.15 0.000 . 7467532 1.034053 -------------+---------------------------------------------------------------- SIGMA2 | _cons | 1.529284 . 1754864 8.71 0.000 1.185337 1.873232 AIC=2.43; SBC=1.25;DW=1.87. Tests Statistic p value Auto Regression (F) 1.78 0.13 ARCH (F) 0.26 0.22 Normality (2 ) 0.53 0.98 Reset (F) 1.11 0.40 4.GARCH MODELS ARCH(1,1) ARCH family regression Sample: 1 to 56 Number of obs = 56 Wald chi2(1) = 0.91 Log likelihood = -77.36367 Prob > chi2 = 0.3404 ------------------------------------------------------------------------------ | OPG dinflation | Coef.

Std. Err. z P>|z| [95% Conf.

Interval] -------------+---------------------------------------------------------------- dinflation | time | -. 0073326 . 0076907 -0.95 0.340 -. 0224061 . 0077409 _cons | . 1759082 . 2722948 0.65 0.518 -. 3577798 . 7095963 -------------+---------------------------------------------------------------- ARCH | arch | L1.

| . 4727264 . 1681697 2.81 0.005 . 1431197 . 802333 garch | L1. | . 5817858 . 0833268 6.98 0.000 . 4184683 . 7451034 _cons | . 0738059 . 1015265 0.73 0.467 -. 1251824 . 2727943 AIC=167.73;SBC=174.85 ARCH(2,2) ARCH family regression Sample: 1 to 56 Number of obs = 56 Wald chi2(1) = 4.56 Log likelihood = -78.17367 Prob > chi2 = 0.02 ------------------------------------------------------------------------------ | OPG dinflation | Coef.

Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- dinflation | time | -. 0120337 . 0074222 -1.62 0.105 -. 026581 . 0025136 _cons | . 3831668 . 2276986 1.68 0.092 -. 0631143 . 8294479 -------------+---------------------------------------------------------------- ARCH | arch | L1.

| . 1750954 . 2314547 0.76 0.449 -. 2785474 . 6287382 L2.

| . 7336057 . 5061762 1.45 0.147 -. 2584813 1.725693 garch | L1. | . 1273698 . 3158046 0.40 0.687 -. 491596 . 7463355 _cons | . 2918694 . 2633994 1.11 0.268 -. 2243839 . 8081226 ------------------------------------------------------------------------------ AIC=162.73; SBC=170.85 5.

VAR Vector autoregression Sample: 3 59 No. of obs = 57 Log likelihood = -424.2599 AIC = 15.23719 FPE = 14219.08 HQIC = 15.37649 Det(Sigma_ml) = 10002.18 SBIC = 15.59562 Equation Parms RMSE R-sq chi2 P>chi2 ---------------------------------------------------------------- inflation 5 . 993154 0.8255 269.6309 0.0000 realgdpus 5 118.755 0.9988 48926.18 0.0000 ---------------------------------------------------------------- ------------------------------------------------------------------------------ | Coef.

Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- inflation | inflation | L1.

| 1.094417 . 1064635 10.28 0.000 . 8857521 1.303081 L2.

| -. 2315049 . 1083356 -2.14 0.033 -. 4438388 -. 0191711 realgdpus | L1. | . 0017795 . 0012138 1.47 0.143 -. 0005995 . 0041585 L2. | -. 0018808 . 001244 -1.51 0.131 -. 0043189 . 0005573 _cons | . 7571558 . 3744524 2.02 0.043 . 0232426 1.491069 -------------+---------------------------------------------------------------- realgdpus inflation | L1.

| -7.862507 12.73027 -0.62 0.537 -32.81338 17.08837 L2. | -5.907147 12.95412 -0.46 0.648 -31.29676 19.48246 realgdpus | L1. | 1.238267 . 1451391 8.53 0.000 . 9537992 1.522734 L2.

| -. 2235117 . 1487448 -1.50 0.133 -. 5150462 . 0680227 _cons | 104.6265 44.77478 2.34 0.019 16.86952 192.3834 ------------------------------------------------------------------------------ Reference Hamilton JD (1994): “Time Series Econometrics”, Princeton, NJ: Princeton University Press.

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