applied econometrics assigment 2

ECON 3P95

Assignment 1

Chenguang Li , Angela Ndlovu , Xingbin Tan

February 12, 2016

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Contents

1 Stationarity for all three variables 41.1 Stationarity for inflation . . . . . . . . . . . . . . . . . . . . . . . 41.2 Stationarity for civilian unemployment rate . . . . . . . . . . . . 51.3 Stationarity for interest rate . . . . . . . . . . . . . . . . . . . . . 6

2 Determine the optimal lag length for the VAR 7

3 Estimate the VAR model with the optimal lag choice 83.1 Dependent variable: inflation . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Coefficients & Fit . . . . . . . . . . . . . . . . . . . . . . . 83.1.2 Graph fitted, actual and residuals . . . . . . . . . . . . . 93.1.3 Correlogram of residuals . . . . . . . . . . . . . . . . . . . 103.1.4 LB test for white noise (Appendix 2.1) . . . . . . . . . . . 10

3.2 Dependent variable: CUR . . . . . . . . . . . . . . . . . . . . . . 113.2.1 Coefficients & Fit . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Graph fitted, actual and residuals . . . . . . . . . . . . . 123.2.3 Correlogram of residuals . . . . . . . . . . . . . . . . . . . 133.2.4 LB test for white noise (Appendix 2.2) . . . . . . . . . . . 13

3.3 Dependent variable: EFFR . . . . . . . . . . . . . . . . . . . . . 143.3.1 Coefficients & Fit . . . . . . . . . . . . . . . . . . . . . . . 143.3.2 Graph fitted, actual and residuals . . . . . . . . . . . . . 153.3.3 Correlogram of residuals . . . . . . . . . . . . . . . . . . . 163.3.4 LB test for white noise (Appendix 2.3) . . . . . . . . . . . 16

4 Granger causality tests for each equation 174.1 Inflation as dependent variable . . . . . . . . . . . . . . . . . . . 17

4.1.1 omit EFFR 1 EFFR 2 . . . . . . . . . . . . . . . . . . . . 174.1.2 omit CUR 1 CUR 2 . . . . . . . . . . . . . . . . . . . . . 18

4.2 CUR as dependent variable . . . . . . . . . . . . . . . . . . . . . 194.2.1 omit inflation 1 inflation 2 . . . . . . . . . . . . . . . . . . 194.2.2 omit EFFR 1 EFFR 2 . . . . . . . . . . . . . . . . . . . . 20

4.3 EFFR as dependent variable . . . . . . . . . . . . . . . . . . . . 214.3.1 omit inflation 1 inflation 2 . . . . . . . . . . . . . . . . . . 214.3.2 omit CUR 1 CUR 2 . . . . . . . . . . . . . . . . . . . . . 22

5 Vector Autoregression (for forecasting 2015) 235.1 Forecasting 2015 CUR . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Forecasting 2015 EFFR . . . . . . . . . . . . . . . . . . . . . . . 245.3 Forecasting 2015 inflation . . . . . . . . . . . . . . . . . . . . . . 25

6 Vector Autoregression (for forecasting 2016) 266.1 Forecasting 2016 CUR . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Forecasting 2016 EFFR . . . . . . . . . . . . . . . . . . . . . . . 276.3 Forecasting 2016 inflation . . . . . . . . . . . . . . . . . . . . . . 28

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7 Impulse Response 29

8 Appendix 318.1 Autocorrelation function for inflation . . . . . . . . . . . . . . . . 318.2 Autocorrelation function for CUR . . . . . . . . . . . . . . . . . 328.3 Autocorrelation function for EFFR . . . . . . . . . . . . . . . . . 338.4 Gretl code for granger causality tests . . . . . . . . . . . . . . . . 348.5 Gretl outout: Vector Autoregression (for forecasting 2015) . . . . 358.6 Gretl output: Forecasting 2015 Forecast evaluation statistics . . 388.7 Gretl outout: Vector Autoregression (for forecasting 2016) . . . . 418.8 Gretl: command log . . . . . . . . . . . . . . . . . . . . . . . . . 44

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1 Stationarity for all three variables

1.1 Stationarity for inflation

Figure 1: correlagram of inflation

From the correlogram we see a decay in the ACF where it starts high, andgradually tappers off, not by large margins and also does not fall within theconfidence interval. The PACF falls within the confidence interval, much closerto zero, which is an indication of stationarity. The LB test in Appendix 1.1,shows high Q-stats, and low p-values f 0.000, which are significant at 1%. Wecan therefore, safely reject the null hypothesis, and conclude no white noise.

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1.2 Stationarity for civilian unemployment rate

Figure 2: correlagram of civilian unemployment rate

From the correlogram we see an obvious decay in the ACF where it starts highand gradually tappers off, eventually falling into the confidence interval. ThePACF also quickly jumps to zero which again is an indication of stationarity.The LB test in Appendix 1.2, shows high Q-stats, and low p-values of 0.000,which are significant at 1%. We can therefore, safely reject the null hypothesis,and conclude no white noise

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1.3 Stationarity for interest rate

Figure 3: correlagram of effective federal funds rate

From the correlogram we see a decay in the ACF where it starts high, andgradually tappers off, not by large margins and also does not fall within theconfidence interval. The PACF also jumps quickly to zero which is an indicationof stationarity. The LB test in Appendix 1.3, shows high Q-stats, and low p-values of 0.000, which are significant at 1%. We can therefore, safely reject thenull hypothesis, and conclude no white noise.

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2 Determine the optimal lag length for the VAR

VAR system, maximum lag order 12The asterisks below indicate the best (that is, minimized) values of the re-

spective information criteria, AIC = Akaike criterion, BIC = Schwarz Bayesiancriterion and HQC = Hannan-Quinn criterion.

Table 1: VAR system, maximum lag order 12lags loglike p(LR) AIC BIC HQC1 -605.28494 5.851042 6.041669 5.9280972 -534.25556 0.00000 5.263086 5.596683* 5.3979323 -514.96166 0.00001 5.165509 5.642076 5.358147*4 -508.04481 0.12840 5.185259 5.804797 5.4356885 -497.76610 0.01477 5.173138 5.935646 5.4813596 -483.70261 0.00091 5.125143 6.030621 5.4911567 -474.97817 0.04213 5.127755 6.176204 5.5515598 -459.92615 0.00042 5.070390 6.261809 5.5519859 -445.80164 0.00087 2.021817* 6.356206 5.561203

The best values of the respective information criterion is BIC at lag 2, there-fore our optimal lag length for VAR is lag 2.

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3 Estimate the VAR model with the optimal lagchoice

3.1 Dependent variable: inflation

Model 1: OLS, using observations 1960:4–2015:4 (T = 221)Dependent variable: inflation

Coefficient Std. Error t-ratio p-value

const 0.437066 0.287570 1.5199 0.1300CUR 1 −0.512652 0.239730 −2.1385 0.0336CUR 2 0.467956 0.234976 1.9915 0.0477EFFR 1 0.131575 0.0869738 1.5128 0.1318EFFR 2 −0.102350 0.0854516 −1.1977 0.2323inflation 1 0.645608 0.0667109 9.6777 0.0000inflation 2 0.257693 0.0689865 3.7354 0.0002

Mean dependent var 3.326106 S.D. dependent var 2.339337Sum squared resid 208.2592 S.E. of regression 0.986496R2 0.827020 Adjusted R2 0.822170F (6, 214) 170.5229 P-value(F ) 1.18e–78Log-likelihood −307.0240 Akaike criterion 628.0480Schwarz criterion 651.8352 Hannan–Quinn 637.6528ρ̂ −0.031481 Durbin’s h −3.646354

3.1.1 Coefficients & Fit

Based on the model above we see that the lags of both unemployment (CUR)and inflation are both significant. The latter being significant at 1% and theformer at 5%, whereas interest rate (EFFE) is not significant based on the highp-value. The adjusted r-squared is very high which is a good thing, as it meansabout 82% of the variations in inflation are explained by both unemploymentand interest rate. The standard error of regression is pretty low and so is thep-value for the f-test showing significance

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3.1.2 Graph fitted, actual and residuals

Figure 4: Fitted, actual and residuals of inflation as dependent variable

There is a pretty good fit between the actual and fitted variables. Residuals arewhite noise, as there are no patterns and have a lot of fluctuations

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3.1.3 Correlogram of residuals

Figure 5: ACF for residual from inflation as dependent variable

Looking at the correlogram of residuals almost all the lags of ACF apart fromlag 2 and 25 fall within the confidence interval. Therefore, if we are not beingstrict we can assume the presence of white noise.

3.1.4 LB test for white noise (Appendix 2.1)

Performing the LB test, (Appendix 2.1) we see high p-values and low Q-statstherefore, we fail to reject the null hypothesis, proving presence of white noisein the residuals

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3.2 Dependent variable: CUR

Model 1: OLS, using observations 1960:4–2015:4 (T = 221)Dependent variable: CUR


const 0.126459 0.0701209 1.8034 0.0727EFFR 1 0.00772826 0.0212077 0.3644 0.7159EFFR 2 0.00476731 0.0208365 0.2288 0.8192inflation 1 0.0226420 0.0162668 1.3919 0.1654inflation 2 −0.0165301 0.0168217 −0.9827 0.3269CUR 1 1.62183 0.0584556 27.7447 0.0000CUR 2 −0.657082 0.0572966 −11.4681 0.0000

Mean dependent var 6.103167 S.D. dependent var 1.592579Sum squared resid 12.38263 S.E. of regression 0.240547R2 0.977808 Adjusted R2 0.977186F (6, 214) 1571.550 P-value(F ) 6.2e–174Log-likelihood 4.861016 Akaike criterion 4.277968Schwarz criterion 28.06511 Hannan–Quinn 13.88279ρ̂ −0.033989 Durbin’s h −1.021175


Based on the model above we see that only the lags of unemployment (CUR)are significant at the 1% level. The adjusted r-squared is very high at 0.9771,meaning about 98% of the variations in unemployment are explained by bothinflation and interest rate. The standard error of regression is pretty low andso is the p-value for the f-test showing significance

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Figure 6: Fitted, actual and residuals of unemployment rate as dependent vari-able

There is a pretty good fit between the actual and fitted variables. Residuals arewhite noise, as there are no patterns and have a lot of fluctuations.

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Figure 7: ACF for residual from unemployment rate as dependent variable

The correlogram of residuals shows almost all the lags of ACF falling within theconfidence interval with a slight exception of lag 7 and 8, though nothing toosignificant to rule out the presence of white noise.


Performing the LB test, (Appendix 2.2) we see high p-values and low Q-statstherefore, we fail to reject the null hypothesis, proving presence of white noisein the residuals.

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3.3 Dependent variable: EFFR

Model 2: OLS, using observations 1960:4–2015:4 (T = 221)Dependent variable: EFFR


const 0.281657 0.244136 1.1537 0.2499inflation 1 0.0150816 0.0566351 0.2663 0.7903inflation 2 0.130771 0.0585670 2.2329 0.0266CUR 1 −0.930319 0.203522 −4.5711 0.0000CUR 2 0.879065 0.199486 4.4066 0.0000EFFR 1 1.01383 0.0738375 13.7306 0.0000EFFR 2 −0.101867 0.0725452 −1.4042 0.1617

Mean dependent var 5.278265 S.D. dependent var 3.651191Sum squared resid 150.1001 S.E. of regression 0.837498R2 0.948821 Adjusted R2 0.947386F (6, 214) 661.2376 P-value(F ) 4.0e–135Log-likelihood −270.8374 Akaike criterion 555.6747Schwarz criterion 579.4619 Hannan–Quinn 565.2796ρ̂ 0.065709 Durbin–Watson 1.866807


Based on the model above we see that the second lag of inflation is significantat 5% whereas the first lag is not. Both lags of unemployment and only thefirst lag of interest rate are significant at the 1% level of significance. However,the second lag of interest rate is not. The adjusted r-squared is very high at0.947386, meaning about 95% of the variations in interest rate are explained byboth unemployment and inflation. The standard error of regression is prettylow and so is the p-value for the f-test showing significance.

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Figure 8: Fitted, actual and residuals of interest rate as dependent variable

There is a pretty good fit between the actual and fitted variables. Residualsshow some fluctuations and no patterns which is therefore, an indication ofwhite noise.

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Figure 9: ACF for residual from interest rate as dependent variable

The correlogram shows a fair amount of the lags of ACF falling within theconfidence interval with an exception of lag 2, 5 and 7. Again nothing toosignificant to rule out the presence of white noise.


Compared to the output for unemployment and inflation, the p-values here arevery low at 0.000, showing significance at the 1% level, also the Q-stats aremuch higher, hence leading to rejecting the null hypothesis which says there iswhite noise. We therefore conclude there is no presence of white noise in theresiduals. Based on these results we can conclude there are still some dynamicsfor the model to catch.

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4 Granger causality tests for each equation

4.1 Inflation as dependent variable

4.1.1 omit EFFR 1 EFFR 2

Null hypothesis: the regression parameters are zero for the variables EFFR 1,EFFR 2Test statistic: F(2, 214) = 1.45951 , p-value 0.234654Omitting variables improved 3 of 3 information criteria.

Table 2: OLS, using observations 1960:4–2015:4 (T = 221)Dependent variable: inflation

coefficient std. error t-ratio p-valueconst 0.535728 0.279994 1.913 0.0570 *inflation 1 0.668745 0.0653064 10.24 2.58e-020 ***inflation 2 0.277192 0.0665731 4.164 4.52e-05 ***CUR 1 -0.673048 0.207994 -3.236 0.0014 ***CUR 2 0.613980 0.206752 2.970 0.0033 ***

Mean dependent var 3.326106 S.D. dependent var 2.339337Sum squared resid 211.0999 S.E. of regression 0.988592R2 0.824660 Adjusted R2 0.821413F (6, 214) 253.9740 P-value(F ) 1.97e–80Log-likelihood −308.5211 Akaike criterion 627.0421Schwarz criterion 644.0330 Hannan–Quinn 633.9027rho 0.035359 Durbin h −2.193078

Based on the high p-value we fail to reject the null hypothesis at 5%, con-cluding that lags of unemployment have no effect on the inflation.

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4.1.2 omit CUR 1 CUR 2




const 0.232886 0.125572 1.855 0.0650 *inflation 1 0.652778 0.0670585 9.734 8.45e-019 ***inflation 2 0.226822 0.0679664 3.337 0.0010 ***EFFR 1 0.228054 0.0758171 3.008 0.0029 ***EFFR 2 −0.196578 0.0744599 −2.640 0.0089 ***

Mean dependent var 3.326106 S.D. dependent var 2.339337Sum squared resid 213.0854 S.E. of regression 0.993230R2 0.823011 Adjusted R2 0.819734F (4, 216) 251.1043 P-value(F ) 5.39e–80Log-likelihood −309.5555 Akaike criterion 629.1110Schwarz criterion 646.1018 Hannan–Quinn 635.9716rho -0.045618 Durbin’s h -8.613685

Based on the p-value we fail to reject the null hypothesis at 5%, but canreject it at the 10% level of significance, concluding that lags of interest ratehave some effect on the inflation. Which holds true as the lower the interest rateis consumers invest less and spend more. Therefore , this increases the economygrowth and in turn increased inflation .

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4.2 CUR as dependent variable

4.2.1 omit inflation 1 inflation 2

Null hypothesis: the regression parameters are zero for the variables inflation 1,inflation 2Test statistic: F(2, 214) = 0.989351 , p-value 0.373512Omitting variables improved 3 of 3 information criteria.



const 0.135683 0.0698086 1.944 0.0532 *EFFR 1 0.0132878 0.0201581 0.6592 0.5105EFFR 2 0.00175603 0.0206988 0.08484 0.9325 ***CUR 1 1.62167 0.0758171 28.68 1.60e-075 ***CUR 2 −0.62167 0.0558229 −11.77 4.77e-025 ***

Mean dependent var 6.103167 S.D. dependent var 1.592579Sum squared resid 12.49712 S.E. of regression 0.240535R2 0.977603 Adjusted R2 0.977188F (4, 216) 2357.062 P-value(F ) 7.0e–177Log-likelihood 3.843998 Akaike criterion 2.312004Schwarz criterion 19.30282 Hannan–Quinn 9.172590rho -0.029104 Durbin’s h 0.800612

Based on the high p-value we fail to reject the null hypothesis at 5%, con-cluding that lags of interest rate have no effect on the unemployment.

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4.2.2 omit EFFR 1 EFFR 2




const 0.156490 0.0683936 2.288 0.0231 **inflation 1 0.0282363 0.0159523 1.770 0.0781 *inflation 2 −0.00782899 0.0162617 −0.4814 0.6307CUR 1 1.62170 0.0508061 31.92 1.01e083 ***CUR 2 −0.628827 0.0505028 −13.05 4.54e-029 ***

Mean dependent var 6.103167 S.D. dependent var 1.592579Sum squared resid 12.59565 S.E. of regression 0.241481R2 0.977427 Adjusted R2 0.977009F (4, 216) 2338.202 P-value(F ) 1.6e–176Log-likelihood 2.976217 Akaike criterion 4.047566Schwarz criterion 21.03838 Hannan–Quinn 21.03838rho − 0.028641 Durbin’s h 0.649654

Based on the p-value of we fail to reject the null hypothesis at 5%, concludingthat lags of inflation have no effect on the unemployment.

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4.3 EFFR as dependent variable

4.3.1 omit inflation 1 inflation 2

Null hypothesis: the regression parameters are zero for the variables inflation 1,inflation 2Test statistic: F(2, 214) = 8.1406 , p-value 0.000391395Omitting variables improved 0 of 3 information criteria.



const 0.294447 0.250967 1.173 0.2420EFFR 1 1.09783 0.0724698 15.150 8.38e-036 ***EFFR 2 −0.730012 0.0744136 −1.647 0.1011CUR 1 −0.730012 0.203475 −3.588 0.0004 ***CUR 2 0.701435 0.200688 3.495 0.0006 ***

Mean dependent var 5.278265 S.D. dependent var 3.651191Sum squared resid 161.5198 S.E. of regression 0.864741R2 0.944928 Adjusted R2 0.943908F (4, 216) 926.5273 P-value(F ) 1.1e–134Log-likelihood −278.9398 Akaike criterion 567.8796Schwarz criterion 584.8704 Hannan–Quinn 574.7402rho 0.049727 Durbin-Watson 1.899101

Based on the low p-value we reject the null hypothesis at 5%, concludingthat lags of unemployment have strong effects on the interest rate

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4.3.2 omit CUR 1 CUR 2

Null hypothesis: the regression parameters are zero for the variables CUR 1,CUR 2Test statistic: F(2, 214) = 10.5714 , p-value 4.18601e-005Omitting variables improved 0 of 3 information criteria.

Model 7: OLS, using observations 1960:4–2015:4 (T = 221)Dependent variable: EFFR


const 0.0933075 0.110475 0.8446 0.3993inflation 1 0.0261197 0.0589965 0.4427 0.6584inflation 2 0.0796156 0.0597953 1.331 0.1844EFFR 1 1.18317 0.0667021 17.740 4.88e-044 ***EFFR 2 −0.268938 0.0655081 4.105 5.72e-05 ***

Mean dependent var 5.278265 S.D. dependent var 3.651191Sum squared resid 164.9297 S.E. of regression 0.873821R2 0.943765 Adjusted R2 0.942724F (4, 216) 906.2551 P-value(F ) 1.0e–133Log-likelihood −281.2483 Akaike criterion 572.4966Schwarz criterion 589.4874 Hannan–Quinn 579.3572rho 0.053325 Durbin’s h 6.128416

Based on the low p-value we again reject the null hypothesis at 5%, conclud-ing that lags of inflation have strong effects on the interest rate

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5 Vector Autoregression (for forecasting 2015)

5.1 Forecasting 2015 CUR

Figure 10: forecasts for civilian unemployment rate 2015

As you can see, visually we have a good forecasts for the downward slopingtrend for unemployment rate and they are all within the 95% confidence interval.However we predict a little too much for the amount of decreasing as the actualunemployment rate was dropping in a smaller pace. We also can find prove ofa good overall forecasts in gretl outputs (see appendix 8.6). The standard errorare all pretty low and also the ME, MSE and MAE are all fairly small. Thatindicates a good over forecasts

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5.2 Forecasting 2015 EFFR

Figure 11: forecasts for effective federal funds rate 2015

As you can see, visually we have obvious differences between the forecasts andthe actual interest rate. Even though they are all within the 95% confidenceinterval, we still predicted upward trend in interest rate while the actual interestrate was more flat. From gretl outputs (see appendix 8.6) we can see a largerstandard error between forecasts and actual value of interest rate. Also the ME,MSE and MAE are rather big compare to our forecasts for unemployment rate.Therefore the forecast for interest rate is a not so good but fairly ok forecasts.

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5.3 Forecasting 2015 inflation

Figure 12: forecasts for inflation 2015

As you can see, visually we have an completely opposite trend between theforecasts and the actual inflation. Even though they are all within the 95%confidence interval, we still predicted upward trend in interest rate while theactual interest rate was decreasing. From gretl outputs (see appendix 8.6) wecan see a similar level of stander error compare to unemployment forecasts. Asfor the ME, MSE and MAE are fairly low. This indicates even we have anopposite of prediction of the trend, but we still had a right prediction on thelevels of the inflation. Therefore, overall we would say forecasts for inflation isa good forecasts.

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6 Vector Autoregression (for forecasting 2016)

6.1 Forecasting 2016 CUR

Figure 13: forecasts for civilian unemployment rate 2016

From November 1982 to July 1990 the U.S. economy experienced robustgrowth and modest unemployment at 5.2% in 1990.” The recession only lastedeight months, however improvements happened slowly, with unemploymentreaching about 8% in 1992. Unemployment continued declining steadily tillabout 2001, where it had reached 4%.The September 11 terrorist attacks could have been a contributing factor in theearly 2000 recession, as we see unemployment increasing again after a ten yeargrowth of the economy. In about 2004 unemployment decreased till the greatdepression of 2008, we again, see a sharp increase in unemployment which sawunemployment reaching highs of 10% in 2010, which is when the depression wasunder control, resulting in a fast decline.[1]Based on the 2016 forecast we observe a continuing declining trend which makessense as it’s expected to keep declining.

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6.2 Forecasting 2016 EFFR

Figure 14: forecasts for effective federal funds rate 2016

As mentioned earlier, there was a recession in the early 1990. It is said thatother causes of that recession where the moves by the U.S. Federal Reserve toraise interest rates in the late 1980s and also Iraq’s invasion of Kuwait in thesummer of 1990. This is visually evident as interest rates in 1990 were peaking8%. [2]The recession was however short lived, hence seeing a sharp decline over thenext 3 years with interest rates stable at 3% in 1993- 1994. They increasedto about 6% by 1995 and dropped sharply in 2001, which was the time of theearly 2000 recession. Interest rates dropped again to about 0% by 2008 at thebeginning of the great depression and have been steady since.The forecast of 2016 over shoots the actual trend observed even though it’swithin the 95% interval.

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6.3 Forecasting 2016 inflation

Figure 15: forecasts for inflation 2016

In the late 1980’s to the early 1990’s the US economy experienced robustgrowth which resulted in low inflation at about 4.5% in the year 1990. Inflationhas been very unstable in the US economy increasing and decreasing randomly.It dropped to below 0% in 2009 which was when the great depression came to anend. It however, started increasing again then had a period of random increaseand decrease between 2011 and 2015.The forecast for 2016 shows an increase in inflation even though 2015 inflationwas at almost 1% 2016 forecast showing an increase to about 2.2%, which couldbe possible based on the pattern it has been following over the years. It is alsocontained within the 95% level confidence interval

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7 Impulse Response

Figure 16: 9 impulse response functions

CUR⇒ CUR: If you shock unemployment on itself it increases to about 0.5by the 4th quarter and gradually but swiftly tappers off to about -0.1 by the19th quarter.

CUR ⇒ EFFR: If you shock interest rate on unemployment, the opposite istrue. It starts at about -0.3 decreases to -0.8 by the 4th quarter then increasesswiftly to almost where it started off, with it being -0.2 by the 19th quarter.

CUR ⇒ INFLATION: The effects of shocking inflation on unemploymentmake sense as an increase in inflation results in lower unemployment. As seenin the figure. Unemployment starts at 0, decreases to -0.3 by the 5th quarterthen gradually starts to raise again to -0.05 by the 19th quarter.

EFFR ⇒ CUR: Shocking unemployment on interest rate we see unemploy-ment starting at 0 then gradually increasing to 0.14 by the 10th quarter. Itslowly starts to decrease to 0.09 by the 19th quarter.

EFFR ⇒ EFFR: Shocking interest rate on itself we see a gradual declinefrom about 0.8 to 0.1 in the 19th quarter.

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EFFR ⇒ INFLATION : Shocking inflation on interest rate we see a sharpincrease in the interest rate from 0 to almost 0.1 with the 1st quarter. There’s asharp drop to 0.08 by the 3rd quarter where it stays constant till the 5th quar-ter, then declines steadily and is back to 0 by the 15th quarter and maintainsthat till the 19th quarter.

INFLATION ⇒ CUR: Shocking unemployment on inflation we see a steadyincrease from 0 to almost 0.2 by the 19th quarter.

INFLATION ⇒ EFFR: Shocking interest rate on inflation starts at 0.1 tillthe 1st quarter and increases to about 0.5 by the 9th quarter. It then graduallydeclines to 0.38 by the 19th quarter.

INFLATION ⇒ INFLATION: When inflation is shocked on itself, it startsat 1 then drops sharply to about 0.67 by the 1st quarter. It increases slightlyto 0.69 in the 2nd quarter then gradually declines to 0.2 in the 19th quarter.

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8 Appendix

8.1 Autocorrelation function for inflation

LAG ACF PACF Q-stat. [p-value]

1 0.8941 *** 0.8941 *** 180.6732 [0.000]

2 0.8434 *** 0.2196 *** 342.1836 [0.000]

3 0.8170 *** 0.1682 ** 494.3990 [0.000]

4 0.7814 *** 0.0264 634.2784 [0.000]

5 0.7202 *** -0.1332 ** 753.6718 [0.000]

6 0.6856 *** 0.0330 862.3691 [0.000]

7 0.6401 *** -0.0622 957.5360 [0.000]

8 0.6087 *** 0.0567 1043.9944 [0.000]

9 0.5779 *** 0.0279 1122.3094 [0.000]

10 0.5650 *** 0.0962 1197.5011 [0.000]

11 0.5373 *** -0.0120 1265.8259 [0.000]

12 0.5185 *** 0.0118 1329.7555 [0.000]

13 0.5118 *** 0.0642 1392.3389 [0.000]

14 0.5083 *** 0.0420 1454.3678 [0.000]

15 0.4956 *** 0.0149 1513.6238 [0.000]

16 0.4736 *** -0.0772 1567.9985 [0.000]

17 0.4604 *** -0.0001 1619.6258 [0.000]

18 0.4442 *** -0.0255 1667.9150 [0.000]

19 0.4334 *** 0.0443 1714.1142 [0.000]

20 0.4199 *** 0.0163 1757.6995 [0.000]

21 0.4073 *** 0.0140 1798.9017 [0.000]

22 0.3971 *** 0.0293 1838.2620 [0.000]

23 0.4043 *** 0.0852 1879.2728 [0.000]

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8.2 Autocorrelation function for CUR


1 0.9759 *** 0.9759 *** 216.2007 [0.000]

2 0.9228 *** -0.6200 *** 410.4029 [0.000]

3 0.8522 *** -0.0566 576.7793 [0.000]

4 0.7731 *** 0.0167 714.3171 [0.000]

5 0.6935 *** 0.0693 825.5048 [0.000]

6 0.6171 *** -0.0240 913.9297 [0.000]

7 0.5406 *** -0.1850 *** 982.0967 [0.000]

8 0.4657 *** 0.0401 1032.9155 [0.000]

9 0.3966 *** 0.1201 * 1069.9594 [0.000]

10 0.3333 *** -0.0556 1096.2394 [0.000]

11 0.2751 *** -0.0719 1114.2238 [0.000]

12 0.2239 *** 0.0493 1126.1955 [0.000]

13 0.1806 *** 0.0778 1134.0206 [0.000]

14 0.1439 ** -0.0255 1139.0133 [0.000]

15 0.1137 * -0.0426 1142.1441 [0.000]

16 0.0888 -0.0132 1144.0614 [0.000]

17 0.0693 0.0971 1145.2362 [0.000]

18 0.0534 -0.0587 1145.9362 [0.000]

19 0.0412 -0.0133 1146.3551 [0.000]

20 0.0323 0.0133 1146.6143 [0.000]

21 0.0268 0.0576 1146.7932 [0.000]

22 0.0224 -0.0678 1146.9189 [0.000]

23 0.0199 0.0125 1147.0184 [0.000]

32

8.3 Autocorrelation function for EFFR


1 0.9636 *** 0.9636 *** 210.8029 [0.000]

2 0.9121 *** -0.2314 *** 400.4984 [0.000]

3 0.8675 *** 0.1213 * 572.8788 [0.000]

4 0.8181 *** -0.1527 ** 726.8900 [0.000]

5 0.7633 *** -0.0453 861.5655 [0.000]

6 0.6999 *** -0.1642 ** 975.3282 [0.000]

7 0.6413 *** 0.0944 1071.2644 [0.000]

8 0.6009 *** 0.1733 *** 1155.9005 [0.000]

9 0.5635 *** -0.0467 1230.6687 [0.000]

10 0.5239 *** -0.0040 1295.5917 [0.000]

11 0.4929 *** 0.0765 1353.3450 [0.000]

12 0.4731 *** 0.0575 1406.7915 [0.000]

13 0.4560 *** -0.0578 1456.6702 [0.000]

14 0.4393 *** 0.0296 1503.1935 [0.000]

15 0.4255 *** 0.0499 1547.0543 [0.000]

16 0.4168 *** 0.0129 1589.3409 [0.000]

17 0.4171 *** 0.0955 1631.8921 [0.000]

18 0.4158 *** -0.0279 1674.3765 [0.000]

19 0.4089 *** -0.0178 1715.6582 [0.000]

20 0.4012 *** -0.0493 1755.5995 [0.000]

21 0.3958 *** 0.0376 1794.6686 [0.000]

22 0.3854 *** -0.0817 1831.8856 [0.000]

23 0.3726 *** 0.0533 1866.8539 [0.000]

33

8.4 Gretl code for granger causality tests

ols CUR 0 inflation(-1 to -2) EFFR(-1 to -2) CUR(-1 to -2)

omit inflation_1 inflation_2

ols CUR 0 inflation(-1 to -2) EFFR(-1 to -2) CUR(-1 to -2)

omit EFFR_1 EFFR_2

ols EFFR 0 inflation(-1 to -2) EFFR(-1 to -2) CUR(-1 to -2)

omit inflation_1 inflation_2

ols EFFR 0 inflation(-1 to -2) EFFR(-1 to -2) CUR(-1 to -2)

omit CUR_1 CUR_2

ols inflation 0 inflation(-1 to -2) EFFR(-1 to -2) CUR(-1 to -2)

omit EFFR_1 EFFR_2

ols inflation 0 inflation(-1 to -2) EFFR(-1 to -2) CUR(-1 to -2)

omit CUR_1 CUR_2

34

8.5 Gretl outout: Vector Autoregression (for forecasting2015)

VAR system, lag order 2OLS estimates, observations 1960:4–2014:4 (T = 217)

Log-likelihood = −546.461Determinant of covariance matrix = 0.0308961AIC = 5.2301BIC = 5.5571HQC = 5.3622Portmanteau test: LB(48) = 573.664, df = 414 [0.0000]

Equation 1: CUR


const 0.122459 0.0713234 1.7170 0.0875CURt−1 1.62800 0.0591725 27.5128 0.0000CURt−2 −0.662925 0.0579556 −11.4385 0.0000EFFRt−1 0.00854371 0.0213839 0.3995 0.6899EFFRt−2 0.00436053 0.0209741 0.2079 0.8355inflationt−1 0.0253736 0.0165872 1.5297 0.1276inflationt−2 −0.0195600 0.0171805 −1.1385 0.2562

Mean dependent var 6.118280 S.D. dependent var 1.603038Sum squared resid 12.30472 S.E. of regression 0.242062R2 0.977832 Adjusted R2 0.977198F (6, 210) 1543.838 P-value(F ) 1.1e–170ρ̂ −0.032606 Durbin–Watson 2.032211

F-tests of zero restrictions

All lags of CUR F (2, 210) = 4170.37 [0.0000]All lags of EFFR F (2, 210) = 1.89718 [0.1526]All lags of inflation F (2, 210) = 1.17995 [0.3093]All vars, lag 2 F (3, 210) = 58.3335 [0.0000]

35

Equation 2: EFFR


const 0.311688 0.248588 1.2538 0.2113CURt−1 −0.950592 0.206237 −4.6092 0.0000CURt−2 0.897253 0.201996 4.4419 0.0000EFFRt−1 1.00956 0.0745306 13.5455 0.0000EFFRt−2 −0.100519 0.0731023 −1.3750 0.1706inflationt−1 0.0108904 0.0578122 0.1884 0.8508inflationt−2 0.136338 0.0598803 2.2768 0.0238

Mean dependent var 5.373118 S.D. dependent var 3.616445Sum squared resid 149.4742 S.E. of regression 0.843672R2 0.947089 Adjusted R2 0.945577F (6, 210) 626.4837 P-value(F ) 4.8e–131ρ̂ 0.066791 Durbin–Watson 1.865396



36

Equation 3: inflation






For the system as a whole —Null hypothesis: the longest lag is 1Alternative hypothesis: the longest lag is 2Likelihood ratio test: χ2

9 = 152.117 [0.0000]

37

8.6 Gretl output: Forecasting 2015 Forecast evaluationstatistics

Output1: forecasting for unemployment rate 2015For 95% confidence intervals, t(210, 0.025) = 1.971

Forecasting 2015 CURCUR prediction std. error 95% interval

2015:1 5.566667 5.309577 0.238126 4.840155 - 5.7790002015:2 5.400000 5.014595 0.452934 4.121715 - 5.9074762015:3 5.166667 4.788693 0.645536 3.516131 - 6.0612542015:4 5.000000 4.625052 0.806969 3.034254 - 6.215850

Forecast evaluation statistics

Mean Error 0.34885Mean Squared Error 0.12452Root Mean Squared Error 0.35287Mean Absolute Error 0.34885Mean Percentage Error 6.6425Mean Absolute Percentage Error 6.6425Theil’s U 1.983Bias proportion, UM 0.97734Regression proportion, UR 0.014591Disturbance proportion, UD 0.008067

38

Output2: forecasting for effected federal funds rate 2015For 95% confidence intervals, t(210, 0.025) = 1.971

Table 3: Forecasting 2015 EFFREFFR prediction std. error 95% interval

2015:1 0.110000 0.704119 0.829953 -0.931987 - 2.3402252015:2 0.123333 1.105314 1.260607 -1.379752 - 3.5903802015:3 0.136667 1.487447 1.603811 -1.674186 - 4.6490802015:4 0.160000 1.810838 1.895295 -1.925403 - 5.547080


Mean Error −1.1444Mean Squared Error 1.4668Root Mean Squared Error 1.2111Mean Absolute Error 1.1444Mean Percentage Error −839.11Mean Absolute Percentage Error 839.11Theil’s U 78.886Bias proportion, UM 0.89292Regression proportion, UR 0.10707Disturbance proportion, UD 8.8448e-006

39

Output3: effected forcasting for inflation rate 2015For 95% confidence intervals, t(210, 0.025) = 1.971

Table 4: Forecasting 2015 inflationEFFR prediction std. error 95% interval

2015:1 0.113661 0.892991 0.968780 -1.016789 - 2.8027712015:2 2.095098 1.074964 1.175392 -1.242116 - 3.3920432015:3 1.310701 1.355684 1.370750 -1.346509 - 4.0578772015:4 0.780736 1.569104 1.526165 -1.439463 - 4.577670


Mean Error −0.14814Mean Squared Error 0.56789Root Mean Squared Error 0.75359Mean Absolute Error 0.6582Mean Percentage Error −185.35Mean Absolute Percentage Error 209.69Theil’s U 0.51574Bias proportion, UM 0.038642Regression proportion, UR 0.048866Disturbance proportion, UD 0.91249

40

8.7 Gretl outout: Vector Autoregression (for forecasting2016)

VAR system, lag order 2OLS estimates, observations 1960:4–2014:4 (T = 217)

Log-likelihood = −546.461Determinant of covariance matrix = 0.0308961

AIC = 5.2301BIC = 5.5571HQC = 5.3622

Portmanteau test: LB(48) = 573.664, df = 414 [0.0000]

Equation 1: CUR


const 0.122459 0.0713234 1.7170 0.0875CURt−1 1.62800 0.0591725 27.5128 0.0000CURt−2 −0.662925 0.0579556 −11.4385 0.0000EFFRt−1 0.00854371 0.0213839 0.3995 0.6899EFFRt−2 0.00436053 0.0209741 0.2079 0.8355inflationt−1 0.0253736 0.0165872 1.5297 0.1276inflationt−2 −0.0195600 0.0171805 −1.1385 0.2562




41

Equation 2: EFFR



Mean dependent var 5.373118 S.D. dependent var 3.616445Sum squared resid 149.4742 S.E. of regression 0.843672R2 0.947089 Adjusted R2 0.945577F (6, 210) 626.4837 P-value(F ) 4.8e–131ρ̂ 0.066791 Durbin–Watson 1.865396



42

Equation 3: inflation






For the system as a whole —Null hypothesis: the longest lag is 1

Alternative hypothesis: the longest lag is 2Likelihood ratio test: χ2

9 = 152.117 [0.0000]

43

8.8 Gretl: command log

# Log started 2016-02-10 11:27

# Record of session commands. Please note that this will

# likely require editing if it is to be run as a script.

open "X:3P952chain-weighted price index.gdt"

lags 1 ; GDP

var 12 CUR EFFR inflation --lagselect

# model 1

ols inflation 0 CUR(-1 to -2) EFFR(-1 to -2) inflation(-1 to -2)

series ResidualInflation = $uhat

setinfo ResidualInflation \par --description="Residual from OLS dependent inflation"

# model 1

series FittedInflation = $yhat

gnuplot inflation ResidualInflation FittedInflation --time-series \par --with-lines

# model 2

ols CUR 0 EFFR(-1 to -2) inflation(-1 to -2) CUR(-1 to -2)

series ResidualCUR = $uhat

# model 3

ols CUR 0 EFFR(-1 to -2) inflation(-1 to -2) CUR(-1 to -2)

series FittedCUR = $yhat

# model 4

ols EFFR 0 inflation(-1 to -2) CUR(-1 to -2) EFFR(-1 to -2)

series ResidualEFFR = $uhat

series FittedEFFR = $yhat

setinfo ResidualEFFR --description="residual from OLS dependent EFFR"

setinfo ResidualInflation \par --description="residual from OLS dependent inflation"

gnuplot inflation ResidualInflation FittedInflation --time-series \par --with-lines

gnuplot CUR ResidualCUR FittedCUR --time-series --with-lines

gnuplot EFFR ResidualEFFR FittedEFFR --time-series --with-lines

corrgm ResidualInflation 23

corrgm ResidualCUR 23

corrgm ResidualEFFR 23

smpl 1960:1 2014:4

var 2 CUR EFFR inflation

var 2 CUR EFFR inflation

44

References

[1] The History of Recessions in the United Stateshttp://useconomy.about.com/od/grossdomesticproduct/a/recession histo.htm

[2] A publication of the Board of Governors of the Federal Reserve Systemhttp://www.federalreserve.gov/pf/pf.htm

45

applied econometrics assigment 2

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