Econ 340 Home Work Assignment 5 Department of Economics Randi Martz Problem 1: (a) Explain the four steps of Box-Jenkin's methods of forecasting. Box-Jenkin's method is used to find the best fit of a time series data model to past values of a time series. 1) Identification of the Model - to do this, there are 3 approaches via graphical, adf test, and act test. If variable is non-stationary, then we need to make it stationary. Then identify an accurate AR or MA model. 2) Estimation of the model - Run the estimated equation results via eviews and which includes the equation model of Dp and the lines in which we see lags 3) Diagnostic test of the model - We need to test the estimated error terms/residuals of the model. We need our estimated value to look like white noise. 4) Forecasting - forecast for a specific time frame in the future using the estimated results. (b) Using Box-Jenkin's four-step method, forecast the US quarterly GDP for the second quarter of 2015. Step 1: Graphical - non-stationary Estimating GDP for ADF Null Hypothesis: IP has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic - based on SIC, maxlag=15) Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level t-Statistic Prob.* -1.593004 -3.992283 -3.426494 -3.136480 0.7935 *MacKinnon (1996) one-sided p-values. t-stat 1.59 < all levels of significance. Therefore, we do NOT reject the H0. GDP is a unit root. Estimating GDP for ACF Date: 12/05/17 Time: 22:39 Sample: 1947Q1 2015Q1 Included observations: 273 Autocorrelation Partial Correlation AC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 PAC 0.990 0.979 0.969 0.958 0.948 0.938 0.927 0.917 0.906 0.896 0.886 0.875 0.865 0.854 0.844 0.990 -0.019 -0.008 0.003 0.001 -0.018 -0.001 0.001 -0.009 -0.006 -0.012 -0.006 -0.003 -0.002 0.005 Q-Stat Prob 270.40 536.06 796.99 1053.3 1305.1 1552.2 1794.8 2032.9 2266.6 2495.8 2720.5 2940.8 3156.6 3368.2 3575.5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 However, this shows that the ACF test suggest GDP is not a unit root variable. Turn this into a stationary variable: DIP 300 200 100 0 -100 -200 -300 -400 50 55 60 65 70 75 80 85 90 95 00 05 10 15 Null Hypothesis: DIP has a unit root Exogenous: None Lag Length: 1 (Automatic - based on SIC, maxlag=15) Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level t-Statistic Prob.* -5.245231 -2.573555 -1.942004 -1.615915 0.0000 *MacKinnon (1996) one-sided p-values. Our DIP is not a unit root variable or a stationary variable. Our results from the ADF test show that coefficient of partial auto correlation at lag 1 = .400. THe coefficient of auto partial correlation at lag 2 = .279. Then, after a few lags, the coefficient drops to less than 0. We use the AR model. Date: 12/05/17 Time: 22:51 Sample: 1947Q1 2015Q1 Included observations: 272 Autocorrelation Partial Correlation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 AC PAC 0.400 0.293 0.172 0.173 0.061 0.059 0.055 0.017 0.134 0.101 0.020 -0.042 -0.044 -0.002 0.400 0.159 0.012 0.080 -0.061 0.012 0.031 -0.034 0.157 0.013 -0.092 -0.062 -0.043 0.058 Q-Stat Prob 44.010 67.780 75.990 84.359 85.385 86.346 87.206 87.287 92.353 95.255 95.366 95.860 96.420 96.421 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 How many lags are included in the AR Model? 1, 2, 9, 24 Estimated Results: Dependent Variable: DIP Method: Least Squares Date: 12/05/17 Time: 23:03 Sample (adjusted): 1953Q2 2015Q1 Included observations: 248 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C DIP(-1) DIP(-2) DIP(-9) DIP(-24) 13.56112 0.331531 0.151769 0.130581 0.157243 6.931425 0.062256 0.062357 0.059107 0.059927 1.956470 5.325306 2.433865 2.209220 2.623911 0.0516 0.0000 0.0157 0.0281 0.0092 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) 0.211726 0.198750 60.98499 903758.1 -1368.807 16.31706 0.000000 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat 55.20444 68.13008 11.07909 11.14992 11.10760 2.009237 Est. for Quarter 2 2015 is =16300.64 =16264.1+21.6+0.33*(16264.1-16294.7)+0.14*(16294.7-16205.6)+0.12*(15538.4-15433.7) Est. for Quarter 2 2015 is =16294.5 =16264.1+13.56+0.33*(16264.1-16294.7)+0.15*(16294.7-16205.6)+0.13*(15538.4-15433.7) observation_date GDP (billion $) 1947-01-01 1947-04-01 1947-07-01 1947-10-01 1948-01-01 1948-04-01 1948-07-01 1948-10-01 1949-01-01 1949-04-01 1949-07-01 1949-10-01 1950-01-01 1950-04-01 1950-07-01 1950-10-01 1951-01-01 1951-04-01 1951-07-01 1951-10-01 1952-01-01 1952-04-01 1952-07-01 1952-10-01 1953-01-01 1953-04-01 1953-07-01 1953-10-01 1954-01-01 1954-04-01 1954-07-01 1954-10-01 1955-01-01 1955-04-01 1955-07-01 1955-10-01 1956-01-01 1956-04-01 1956-07-01 1956-10-01 1957-01-01 1957-04-01 1957-07-01 1957-10-01 1958-01-01 1958-04-01 1958-07-01 1958-10-01 1959-01-01 1934.5 1932.3 1930.3 1960.7 1989.5 2021.9 18000.0 2033.2 16000.0 2035.3 14000.0 2007.5 2000.8 12000.0 2022.8 10000.0 2004.7 8000.0 2084.6 6000.0 2147.6 4000.0 2230.4 2000.0 2273.4 0.0 2304.5 2344.5 2392.8 2398.1 2423.5 2428.5 2446.1 2526.4 2573.4 2593.5 2578.9 2539.8 2528.0 2530.7 2559.4 2609.3 2683.8 2727.5 2764.1 2780.8 2770.0 2792.9 2790.6 2836.2 2854.5 2848.2 2875.9 2846.4 2772.7 2790.9 2855.5 2922.3 2976.6 GDP (billion $) GDP ( $) 1959-04-01 1959-07-01 1959-10-01 1960-01-01 1960-04-01 1960-07-01 1960-10-01 1961-01-01 1961-04-01 1961-07-01 1961-10-01 1962-01-01 1962-04-01 1962-07-01 1962-10-01 1963-01-01 1963-04-01 1963-07-01 1963-10-01 1964-01-01 1964-04-01 1964-07-01 1964-10-01 1965-01-01 1965-04-01 1965-07-01 1965-10-01 1966-01-01 1966-04-01 1966-07-01 1966-10-01 1967-01-01 1967-04-01 1967-07-01 1967-10-01 1968-01-01 1968-04-01 1968-07-01 1968-10-01 1969-01-01 1969-04-01 1969-07-01 1969-10-01 1970-01-01 1970-04-01 1970-07-01 1970-10-01 1971-01-01 1971-04-01 1971-07-01 3049.0 3043.1 3055.1 3123.2 3111.3 3119.1 3081.3 3102.3 3159.9 3212.6 3277.7 3336.8 3372.7 3404.8 3418.0 3456.1 3501.1 3569.5 3595.0 3672.7 3716.4 3766.9 3780.2 3873.5 3926.4 4006.2 4100.6 4201.9 4219.1 4249.2 4285.6 4324.9 4328.7 4366.1 4401.2 4490.6 4566.4 4599.3 4619.8 4691.6 4706.7 4736.1 4715.5 4707.1 4715.4 4757.2 4708.3 4834.3 4861.9 4900.0 1971-10-01 1972-01-01 1972-04-01 1972-07-01 1972-10-01 1973-01-01 1973-04-01 1973-07-01 1973-10-01 1974-01-01 1974-04-01 1974-07-01 1974-10-01 1975-01-01 1975-04-01 1975-07-01 1975-10-01 1976-01-01 1976-04-01 1976-07-01 1976-10-01 1977-01-01 1977-04-01 1977-07-01 1977-10-01 1978-01-01 1978-04-01 1978-07-01 1978-10-01 1979-01-01 1979-04-01 1979-07-01 1979-10-01 1980-01-01 1980-04-01 1980-07-01 1980-10-01 1981-01-01 1981-04-01 1981-07-01 1981-10-01 1982-01-01 1982-04-01 1982-07-01 1982-10-01 1983-01-01 1983-04-01 1983-07-01 1983-10-01 1984-01-01 4914.3 5002.4 5118.3 5165.4 5251.2 5380.5 5441.5 5411.9 5462.4 5417.0 5431.3 5378.7 5357.2 5292.4 5333.2 5421.4 5494.4 5618.5 5661.0 5689.8 5732.5 5799.2 5913.0 6017.6 6018.2 6039.2 6274.0 6335.3 6420.3 6433.0 6440.8 6487.1 6503.9 6524.9 6392.6 6382.9 6501.2 6635.7 6587.3 6662.9 6585.1 6475.0 6510.2 6486.8 6493.1 6578.2 6728.3 6860.0 7001.5 7140.6 1984-04-01 1984-07-01 1984-10-01 1985-01-01 1985-04-01 1985-07-01 1985-10-01 1986-01-01 1986-04-01 1986-07-01 1986-10-01 1987-01-01 1987-04-01 1987-07-01 1987-10-01 1988-01-01 1988-04-01 1988-07-01 1988-10-01 1989-01-01 1989-04-01 1989-07-01 1989-10-01 1990-01-01 1990-04-01 1990-07-01 1990-10-01 1991-01-01 1991-04-01 1991-07-01 1991-10-01 1992-01-01 1992-04-01 1992-07-01 1992-10-01 1993-01-01 1993-04-01 1993-07-01 1993-10-01 1994-01-01 1994-04-01 1994-07-01 1994-10-01 1995-01-01 1995-04-01 1995-07-01 1995-10-01 1996-01-01 1996-04-01 1996-07-01 7266.0 7337.5 7396.0 7469.5 7537.9 7655.2 7712.6 7784.1 7819.8 7898.6 7939.5 7995.0 8084.7 8158.0 8292.7 8339.3 8449.5 8498.3 8610.9 8697.7 8766.1 8831.5 8850.2 8947.1 8981.7 8983.9 8907.4 8865.6 8934.4 8977.3 9016.4 9123.0 9223.5 9313.2 9406.5 9424.1 9480.1 9526.3 9653.5 9748.2 9881.4 9939.7 10052.5 10086.9 10122.1 10208.8 10281.2 10348.7 10529.4 10626.8 1996-10-01 1997-01-01 1997-04-01 1997-07-01 1997-10-01 1998-01-01 1998-04-01 1998-07-01 1998-10-01 1999-01-01 1999-04-01 1999-07-01 1999-10-01 2000-01-01 2000-04-01 2000-07-01 2000-10-01 2001-01-01 2001-04-01 2001-07-01 2001-10-01 2002-01-01 2002-04-01 2002-07-01 2002-10-01 2003-01-01 2003-04-01 2003-07-01 2003-10-01 2004-01-01 2004-04-01 2004-07-01 2004-10-01 2005-01-01 2005-04-01 2005-07-01 2005-10-01 2006-01-01 2006-04-01 2006-07-01 2006-10-01 2007-01-01 2007-04-01 2007-07-01 2007-10-01 2008-01-01 2008-04-01 2008-07-01 2008-10-01 2009-01-01 10739.1 10820.9 10984.2 11124.0 11210.3 11321.2 11431.0 11580.6 11770.7 11864.7 11962.5 12113.1 12323.3 12359.1 12592.5 12607.7 12679.3 12643.3 12710.3 12670.1 12705.3 12822.3 12893.0 12955.8 12964.0 13031.2 13152.1 13372.4 13528.7 13606.5 13706.2 13830.8 13950.4 14099.1 14172.7 14291.8 14373.4 14546.1 14589.6 14602.6 14716.9 14726.0 14838.7 14938.5 14991.8 14889.5 14963.4 14891.6 14577.0 14375.0 16300.64 2009-04-01 2009-07-01 2009-10-01 2010-01-01 2010-04-01 2010-07-01 2010-10-01 2011-01-01 2011-04-01 2011-07-01 2011-10-01 2012-01-01 2012-04-01 2012-07-01 2012-10-01 2013-01-01 2013-04-01 2013-07-01 2013-10-01 2014-01-01 2014-04-01 2014-07-01 2014-10-01 2015-01-01 14355.6 14402.5 14541.9 14604.8 14745.9 14845.5 14939.0 14881.3 14989.6 15021.1 15190.3 15275.0 15336.7 15431.3 15433.7 15538.4 15606.6 15779.9 15916.2 15831.7 16010.4 16205.6 16294.7 16264.1 16264 .1+21.60+0. 33(16264 . 116294 billion $) GDP (billion $) 21.60+0 .33(16264 .116294 .7)+0 .14(16294 . 716205 .6)+0. 12(15538 . 415433 .7)