The document analyzes the relationships between inflation, unemployment, and interest rates using a vector autoregression (VAR) model. It finds that: 1) All three variables - inflation, unemployment, and interest rates - are stationary based on tests of their autocorrelation functions. 2) A VAR with a lag length of 2 is optimal based on information criteria. 3) In the estimated VAR models, lags of inflation and unemployment are significant predictors of current inflation, while only lags of unemployment are significant for current unemployment. 4) Diagnostic tests of residuals show white noise, validating the fitted VAR models. Forecasts for 2015-2016 are also generated from the models.

1. ECON 3P95
Assignment 1
Chenguang Li , Angela Ndlovu , Xingbin Tan
February 12, 2016
1

2. Contents
1 Stationarity for all three variables 4
1.1 Stationarity for inflation . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Stationarity for civilian unemployment rate . . . . . . . . . . . . 5
1.3 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 8
3.1 Dependent variable: inflation . . . . . . . . . . . . . . . . . . . . 8
3.1.1 Coefficients & Fit . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.2 Graph fitted, actual and residuals . . . . . . . . . . . . . 9
3.1.3 Correlogram of residuals . . . . . . . . . . . . . . . . . . . 10
3.1.4 LB test for white noise (Appendix 2.1) . . . . . . . . . . . 10
3.2 Dependent variable: CUR . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Coefficients & Fit . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.2 Graph fitted, actual and residuals . . . . . . . . . . . . . 12
3.2.3 Correlogram of residuals . . . . . . . . . . . . . . . . . . . 13
3.2.4 LB test for white noise (Appendix 2.2) . . . . . . . . . . . 13
3.3 Dependent variable: EFFR . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 Coefficients & Fit . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.2 Graph fitted, actual and residuals . . . . . . . . . . . . . 15
3.3.3 Correlogram of residuals . . . . . . . . . . . . . . . . . . . 16
3.3.4 LB test for white noise (Appendix 2.3) . . . . . . . . . . . 16
4 Granger causality tests for each equation 17
4.1 Inflation as dependent variable . . . . . . . . . . . . . . . . . . . 17
4.1.1 omit EFFR 1 EFFR 2 . . . . . . . . . . . . . . . . . . . . 17
4.1.2 omit CUR 1 CUR 2 . . . . . . . . . . . . . . . . . . . . . 18
4.2 CUR as dependent variable . . . . . . . . . . . . . . . . . . . . . 19
4.2.1 omit inflation 1 inflation 2 . . . . . . . . . . . . . . . . . . 19
4.2.2 omit EFFR 1 EFFR 2 . . . . . . . . . . . . . . . . . . . . 20
4.3 EFFR as dependent variable . . . . . . . . . . . . . . . . . . . . 21
4.3.1 omit inflation 1 inflation 2 . . . . . . . . . . . . . . . . . . 21
4.3.2 omit CUR 1 CUR 2 . . . . . . . . . . . . . . . . . . . . . 22
5 Vector Autoregression (for forecasting 2015) 23
5.1 Forecasting 2015 CUR . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Forecasting 2015 EFFR . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Forecasting 2015 inflation . . . . . . . . . . . . . . . . . . . . . . 25
6 Vector Autoregression (for forecasting 2016) 26
6.1 Forecasting 2016 CUR . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2 Forecasting 2016 EFFR . . . . . . . . . . . . . . . . . . . . . . . 27
6.3 Forecasting 2016 inflation . . . . . . . . . . . . . . . . . . . . . . 28
2

3. 7 Impulse Response 29
8 Appendix 31
8.1 Autocorrelation function for inflation . . . . . . . . . . . . . . . . 31
8.2 Autocorrelation function for CUR . . . . . . . . . . . . . . . . . 32
8.3 Autocorrelation function for EFFR . . . . . . . . . . . . . . . . . 33
8.4 Gretl code for granger causality tests . . . . . . . . . . . . . . . . 34
8.5 Gretl outout: Vector Autoregression (for forecasting 2015) . . . . 35
8.6 Gretl output: Forecasting 2015 Forecast evaluation statistics . . 38
8.7 Gretl outout: Vector Autoregression (for forecasting 2016) . . . . 41
8.8 Gretl: command log . . . . . . . . . . . . . . . . . . . . . . . . . 44
3

4. 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, and
gradually tappers off, not by large margins and also does not fall within the
confidence interval. The PACF falls within the confidence interval, much closer
to 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%. We
can therefore, safely reject the null hypothesis, and conclude no white noise.
4

5. 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 high
and gradually tappers off, eventually falling into the confidence interval. The
PACF 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
5

6. 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, and
gradually tappers off, not by large margins and also does not fall within the
confidence interval. The PACF also jumps quickly to zero which is an indication
of 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 the
null hypothesis, and conclude no white noise.
6

7. 2 Determine the optimal lag length for the VAR
VAR system, maximum lag order 12
The asterisks below indicate the best (that is, minimized) values of the re-
spective information criteria, AIC = Akaike criterion, BIC = Schwarz Bayesian
criterion and HQC = Hannan-Quinn criterion.
Table 1: VAR system, maximum lag order 12
lags loglike p(LR) AIC BIC HQC
1 -605.28494 5.851042 6.041669 5.928097
2 -534.25556 0.00000 5.263086 5.596683* 5.397932
3 -514.96166 0.00001 5.165509 5.642076 5.358147*
4 -508.04481 0.12840 5.185259 5.804797 5.435688
5 -497.76610 0.01477 5.173138 5.935646 5.481359
6 -483.70261 0.00091 5.125143 6.030621 5.491156
7 -474.97817 0.04213 5.127755 6.176204 5.551559
8 -459.92615 0.00042 5.070390 6.261809 5.551985
9 -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.
7

8. 3 Estimate the VAR model with the optimal lag
choice
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 0.1300
CUR 1 −0.512652 0.239730 −2.1385 0.0336
CUR 2 0.467956 0.234976 1.9915 0.0477
EFFR 1 0.131575 0.0869738 1.5128 0.1318
EFFR 2 −0.102350 0.0854516 −1.1977 0.2323
inflation 1 0.645608 0.0667109 9.6777 0.0000
inflation 2 0.257693 0.0689865 3.7354 0.0002
Mean dependent var 3.326106 S.D. dependent var 2.339337
Sum squared resid 208.2592 S.E. of regression 0.986496
R2
0.827020 Adjusted R2
0.822170
F(6, 214) 170.5229 P-value(F) 1.18e–78
Log-likelihood −307.0240 Akaike criterion 628.0480
Schwarz criterion 651.8352 Hannan–Quinn 637.6528
ˆρ −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 the
former at 5%, whereas interest rate (EFFE) is not significant based on the high
p-value. The adjusted r-squared is very high which is a good thing, as it means
about 82% of the variations in inflation are explained by both unemployment
and interest rate. The standard error of regression is pretty low and so is the
p-value for the f-test showing significance
8

9. 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 are
white noise, as there are no patterns and have a lot of fluctuations
9

10. 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 from
lag 2 and 25 fall within the confidence interval. Therefore, if we are not being
strict 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-stats
therefore, we fail to reject the null hypothesis, proving presence of white noise
in the residuals
10

11. 3.2 Dependent variable: CUR
Model 1: OLS, using observations 1960:4–2015:4 (T = 221)
Dependent variable: CUR
Coefficient Std. Error t-ratio p-value
const 0.126459 0.0701209 1.8034 0.0727
EFFR 1 0.00772826 0.0212077 0.3644 0.7159
EFFR 2 0.00476731 0.0208365 0.2288 0.8192
inflation 1 0.0226420 0.0162668 1.3919 0.1654
inflation 2 −0.0165301 0.0168217 −0.9827 0.3269
CUR 1 1.62183 0.0584556 27.7447 0.0000
CUR 2 −0.657082 0.0572966 −11.4681 0.0000
Mean dependent var 6.103167 S.D. dependent var 1.592579
Sum squared resid 12.38263 S.E. of regression 0.240547
R2
0.977808 Adjusted R2
0.977186
F(6, 214) 1571.550 P-value(F) 6.2e–174
Log-likelihood 4.861016 Akaike criterion 4.277968
Schwarz criterion 28.06511 Hannan–Quinn 13.88279
ˆρ −0.033989 Durbin’s h −1.021175
3.2.1 Coefficients & Fit
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 both
inflation and interest rate. The standard error of regression is pretty low and
so is the p-value for the f-test showing significance
11

12. 3.2.2 Graph fitted, actual and residuals
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 are
white noise, as there are no patterns and have a lot of fluctuations.
12

13. 3.2.3 Correlogram of residuals
Figure 7: ACF for residual from unemployment rate as dependent variable
The correlogram of residuals shows almost all the lags of ACF falling within the
confidence interval with a slight exception of lag 7 and 8, though nothing too
significant to rule out the presence of white noise.
3.2.4 LB test for white noise (Appendix 2.2)
Performing the LB test, (Appendix 2.2) we see high p-values and low Q-stats
therefore, we fail to reject the null hypothesis, proving presence of white noise
in the residuals.
13

14. 3.3 Dependent variable: EFFR
Model 2: OLS, using observations 1960:4–2015:4 (T = 221)
Dependent variable: EFFR
Coefficient Std. Error t-ratio p-value
const 0.281657 0.244136 1.1537 0.2499
inflation 1 0.0150816 0.0566351 0.2663 0.7903
inflation 2 0.130771 0.0585670 2.2329 0.0266
CUR 1 −0.930319 0.203522 −4.5711 0.0000
CUR 2 0.879065 0.199486 4.4066 0.0000
EFFR 1 1.01383 0.0738375 13.7306 0.0000
EFFR 2 −0.101867 0.0725452 −1.4042 0.1617
Mean dependent var 5.278265 S.D. dependent var 3.651191
Sum squared resid 150.1001 S.E. of regression 0.837498
R2
0.948821 Adjusted R2
0.947386
F(6, 214) 661.2376 P-value(F) 4.0e–135
Log-likelihood −270.8374 Akaike criterion 555.6747
Schwarz criterion 579.4619 Hannan–Quinn 565.2796
ˆρ 0.065709 Durbin–Watson 1.866807
3.3.1 Coefficients & Fit
Based on the model above we see that the second lag of inflation is significant
at 5% whereas the first lag is not. Both lags of unemployment and only the
first 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 at
0.947386, meaning about 95% of the variations in interest rate are explained by
both unemployment and inflation. The standard error of regression is pretty
low and so is the p-value for the f-test showing significance.
14

15. 3.3.2 Graph fitted, actual and residuals
Figure 8: Fitted, actual and residuals of interest rate as dependent variable
There is a pretty good fit between the actual and fitted variables. Residuals
show some fluctuations and no patterns which is therefore, an indication of
white noise.
15

16. 3.3.3 Correlogram of residuals
Figure 9: ACF for residual from interest rate as dependent variable
The correlogram shows a fair amount of the lags of ACF falling within the
confidence interval with an exception of lag 2, 5 and 7. Again nothing too
significant to rule out the presence of white noise.
3.3.4 LB test for white noise (Appendix 2.3)
Compared to the output for unemployment and inflation, the p-values here are
very low at 0.000, showing significance at the 1% level, also the Q-stats are
much higher, hence leading to rejecting the null hypothesis which says there is
white noise. We therefore conclude there is no presence of white noise in the
residuals. Based on these results we can conclude there are still some dynamics
for the model to catch.
16

17. 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 2
Test statistic: F(2, 214) = 1.45951 , p-value 0.234654
Omitting 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-value
const 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 2.339337
Sum squared resid 211.0999 S.E. of regression 0.988592
R2
0.824660 Adjusted R2
0.821413
F(6, 214) 253.9740 P-value(F) 1.97e–80
Log-likelihood −308.5211 Akaike criterion 627.0421
Schwarz criterion 644.0330 Hannan–Quinn 633.9027
rho 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.
17

18. 4.1.2 omit CUR 1 CUR 2
Null hypothesis: the regression parameters are zero for the variables EFFR 1,
EFFR 2
Test statistic: F(2, 214) = 2.47961 , p-value 0.0861802
Omitting variables improved 2 of 3 information criteria.
Model 2: OLS, using observations 1960:4–2015:4 (T = 221)
Dependent variable: inflation
Coefficient Std. Error t-ratio p-value
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 2.339337
Sum squared resid 213.0854 S.E. of regression 0.993230
R2
0.823011 Adjusted R2
0.819734
F(4, 216) 251.1043 P-value(F) 5.39e–80
Log-likelihood −309.5555 Akaike criterion 629.1110
Schwarz criterion 646.1018 Hannan–Quinn 635.9716
rho -0.045618 Durbin’s h -8.613685
Based on the p-value we fail to reject the null hypothesis at 5%, but can
reject it at the 10% level of significance, concluding that lags of interest rate
have some effect on the inflation. Which holds true as the lower the interest rate
is consumers invest less and spend more. Therefore , this increases the economy
growth and in turn increased inflation .
18

19. 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 2
Test statistic: F(2, 214) = 0.989351 , p-value 0.373512
Omitting variables improved 3 of 3 information criteria.
Model 4: OLS, using observations 1960:4–2015:4 (T = 221)
Dependent variable: CUR
Coefficient Std. Error t-ratio p-value
const 0.135683 0.0698086 1.944 0.0532 *
EFFR 1 0.0132878 0.0201581 0.6592 0.5105
EFFR 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 1.592579
Sum squared resid 12.49712 S.E. of regression 0.240535
R2
0.977603 Adjusted R2
0.977188
F(4, 216) 2357.062 P-value(F) 7.0e–177
Log-likelihood 3.843998 Akaike criterion 2.312004
Schwarz criterion 19.30282 Hannan–Quinn 9.172590
rho -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.
19

20. 4.2.2 omit EFFR 1 EFFR 2
Null hypothesis: the regression parameters are zero for the variables EFFR 1,
EFFR 2
Test statistic: F(2, 214) = 1.84075 , p-value 0.161202
Omitting variables improved 3 of 3 information criteria.
Model 5: OLS, using observations 1960:4–2015:4 (T = 221)
Dependent variable: CUR
Coefficient Std. Error t-ratio p-value
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 0.6307
CUR 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 1.592579
Sum squared resid 12.59565 S.E. of regression 0.241481
R2
0.977427 Adjusted R2
0.977009
F(4, 216) 2338.202 P-value(F) 1.6e–176
Log-likelihood 2.976217 Akaike criterion 4.047566
Schwarz criterion 21.03838 Hannan–Quinn 21.03838
rho − 0.028641 Durbin’s h 0.649654
Based on the p-value of we fail to reject the null hypothesis at 5%, concluding
that lags of inflation have no effect on the unemployment.
20

21. 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 2
Test statistic: F(2, 214) = 8.1406 , p-value 0.000391395
Omitting variables improved 0 of 3 information criteria.
Model 6: OLS, using observations 1960:4–2015:4 (T = 221)
Dependent variable: inflation
Coefficient Std. Error t-ratio p-value
const 0.294447 0.250967 1.173 0.2420
EFFR 1 1.09783 0.0724698 15.150 8.38e-036 ***
EFFR 2 −0.730012 0.0744136 −1.647 0.1011
CUR 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 3.651191
Sum squared resid 161.5198 S.E. of regression 0.864741
R2
0.944928 Adjusted R2
0.943908
F(4, 216) 926.5273 P-value(F) 1.1e–134
Log-likelihood −278.9398 Akaike criterion 567.8796
Schwarz criterion 584.8704 Hannan–Quinn 574.7402
rho 0.049727 Durbin-Watson 1.899101
Based on the low p-value we reject the null hypothesis at 5%, concluding
that lags of unemployment have strong effects on the interest rate
21

22. 4.3.2 omit CUR 1 CUR 2
Null hypothesis: the regression parameters are zero for the variables CUR 1,
CUR 2
Test statistic: F(2, 214) = 10.5714 , p-value 4.18601e-005
Omitting variables improved 0 of 3 information criteria.
Model 7: OLS, using observations 1960:4–2015:4 (T = 221)
Dependent variable: EFFR
Coefficient Std. Error t-ratio p-value
const 0.0933075 0.110475 0.8446 0.3993
inflation 1 0.0261197 0.0589965 0.4427 0.6584
inflation 2 0.0796156 0.0597953 1.331 0.1844
EFFR 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 3.651191
Sum squared resid 164.9297 S.E. of regression 0.873821
R2
0.943765 Adjusted R2
0.942724
F(4, 216) 906.2551 P-value(F) 1.0e–133
Log-likelihood −281.2483 Akaike criterion 572.4966
Schwarz criterion 589.4874 Hannan–Quinn 579.3572
rho 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
22

23. 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 sloping
trend 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 actual
unemployment rate was dropping in a smaller pace. We also can find prove of
a good overall forecasts in gretl outputs (see appendix 8.6). The standard error
are all pretty low and also the ME, MSE and MAE are all fairly small. That
indicates a good over forecasts
23

24. 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 and
the actual interest rate. Even though they are all within the 95% confidence
interval, we still predicted upward trend in interest rate while the actual interest
rate was more flat. From gretl outputs (see appendix 8.6) we can see a larger
standard 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.
24

25. 5.3 Forecasting 2015 inflation
Figure 12: forecasts for inflation 2015
As you can see, visually we have an completely opposite trend between the
forecasts and the actual inflation. Even though they are all within the 95%
confidence interval, we still predicted upward trend in interest rate while the
actual interest rate was decreasing. From gretl outputs (see appendix 8.6) we
can see a similar level of stander error compare to unemployment forecasts. As
for the ME, MSE and MAE are fairly low. This indicates even we have an
opposite of prediction of the trend, but we still had a right prediction on the
levels of the inflation. Therefore, overall we would say forecasts for inflation is
a good forecasts.
25

26. 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 robust
growth and modest unemployment at 5.2% in 1990.” The recession only lasted
eight months, however improvements happened slowly, with unemployment
reaching about 8% in 1992. Unemployment continued declining steadily till
about 2001, where it had reached 4%.
The September 11 terrorist attacks could have been a contributing factor in the
early 2000 recession, as we see unemployment increasing again after a ten year
growth of the economy. In about 2004 unemployment decreased till the great
depression of 2008, we again, see a sharp increase in unemployment which saw
unemployment reaching highs of 10% in 2010, which is when the depression was
under control, resulting in a fast decline.[1]
Based on the 2016 forecast we observe a continuing declining trend which makes
sense as it’s expected to keep declining.
26

27. 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 that
other causes of that recession where the moves by the U.S. Federal Reserve to
raise interest rates in the late 1980s and also Iraq’s invasion of Kuwait in the
summer of 1990. This is visually evident as interest rates in 1990 were peaking
8%. [2]
The recession was however short lived, hence seeing a sharp decline over the
next 3 years with interest rates stable at 3% in 1993- 1994. They increased
to about 6% by 1995 and dropped sharply in 2001, which was the time of the
early 2000 recession. Interest rates dropped again to about 0% by 2008 at the
beginning of the great depression and have been steady since.
The forecast of 2016 over shoots the actual trend observed even though it’s
within the 95% interval.
27

28. 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 robust
growth which resulted in low inflation at about 4.5% in the year 1990. Inflation
has 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 an
end. It however, started increasing again then had a period of random increase
and decrease between 2011 and 2015.
The forecast for 2016 shows an increase in inflation even though 2015 inflation
was at almost 1% 2016 forecast showing an increase to about 2.2%, which could
be possible based on the pattern it has been following over the years. It is also
contained within the 95% level confidence interval
28

29. 7 Impulse Response
Figure 16: 9 impulse response functions
CUR ⇒ CUR: If you shock unemployment on itself it increases to about 0.5
by the 4th quarter and gradually but swiftly tappers off to about -0.1 by the
19th quarter.
CUR ⇒ EFFR: If you shock interest rate on unemployment, the opposite is
true. It starts at about -0.3 decreases to -0.8 by the 4th quarter then increases
swiftly to almost where it started off, with it being -0.2 by the 19th quarter.
CUR ⇒ INFLATION: The effects of shocking inflation on unemployment
make sense as an increase in inflation results in lower unemployment. As seen
in the figure. Unemployment starts at 0, decreases to -0.3 by the 5th quarter
then 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. It
slowly starts to decrease to 0.09 by the 19th quarter.
EFFR ⇒ EFFR: Shocking interest rate on itself we see a gradual decline
from about 0.8 to 0.1 in the 19th quarter.
29

30. EFFR ⇒ INFLATION : Shocking inflation on interest rate we see a sharp
increase in the interest rate from 0 to almost 0.1 with the 1st quarter. There’s a
sharp 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 maintains
that till the 19th quarter.
INFLATION ⇒ CUR: Shocking unemployment on inflation we see a steady
increase from 0 to almost 0.2 by the 19th quarter.
INFLATION ⇒ EFFR: Shocking interest rate on inflation starts at 0.1 till
the 1st quarter and increases to about 0.5 by the 9th quarter. It then gradually
declines to 0.38 by the 19th quarter.
INFLATION ⇒ INFLATION: When inflation is shocked on itself, it starts
at 1 then drops sharply to about 0.67 by the 1st quarter. It increases slightly
to 0.69 in the 2nd quarter then gradually declines to 0.2 in the 19th quarter.
30

31. 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]
31

32. 8.2 Autocorrelation function for CUR
LAG ACF PACF Q-stat. [p-value]
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

33. 8.3 Autocorrelation function for EFFR
LAG ACF PACF Q-stat. [p-value]
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

34. 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

35. 8.5 Gretl outout: Vector Autoregression (for forecasting
2015)
VAR system, lag order 2
OLS estimates, observations 1960:4–2014:4 (T = 217)
Log-likelihood = −546.461
Determinant of covariance matrix = 0.0308961
AIC = 5.2301
BIC = 5.5571
HQC = 5.3622
Portmanteau test: LB(48) = 573.664, df = 414 [0.0000]
Equation 1: CUR
Coefficient Std. Error t-ratio p-value
const 0.122459 0.0713234 1.7170 0.0875
CURt−1 1.62800 0.0591725 27.5128 0.0000
CURt−2 −0.662925 0.0579556 −11.4385 0.0000
EFFRt−1 0.00854371 0.0213839 0.3995 0.6899
EFFRt−2 0.00436053 0.0209741 0.2079 0.8355
inflationt−1 0.0253736 0.0165872 1.5297 0.1276
inflationt−2 −0.0195600 0.0171805 −1.1385 0.2562
Mean dependent var 6.118280 S.D. dependent var 1.603038
Sum squared resid 12.30472 S.E. of regression 0.242062
R2
0.977832 Adjusted R2
0.977198
F(6, 210) 1543.838 P-value(F) 1.1e–170
ˆρ −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

36. Equation 2: EFFR
Coefficient Std. Error t-ratio p-value
const 0.311688 0.248588 1.2538 0.2113
CURt−1 −0.950592 0.206237 −4.6092 0.0000
CURt−2 0.897253 0.201996 4.4419 0.0000
EFFRt−1 1.00956 0.0745306 13.5455 0.0000
EFFRt−2 −0.100519 0.0731023 −1.3750 0.1706
inflationt−1 0.0108904 0.0578122 0.1884 0.8508
inflationt−2 0.136338 0.0598803 2.2768 0.0238
Mean dependent var 5.373118 S.D. dependent var 3.616445
Sum squared resid 149.4742 S.E. of regression 0.843672
R2
0.947089 Adjusted R2
0.945577
F(6, 210) 626.4837 P-value(F) 4.8e–131
ˆρ 0.066791 Durbin–Watson 1.865396
F-tests of zero restrictions
All lags of CUR F(2, 210) = 10.7551 [0.0000]
All lags of EFFR F(2, 210) = 788.22 [0.0000]
All lags of inflation F(2, 210) = 8.18794 [0.0004]
All vars, lag 2 F(3, 210) = 12.1922 [0.0000]
36

37. Equation 3: inflation
Coefficient Std. Error t-ratio p-value
const 0.446412 0.290169 1.5385 0.1254
CURt−1 −0.541048 0.240735 −2.2475 0.0256
CURt−2 0.494421 0.235784 2.0969 0.0372
EFFRt−1 0.125446 0.0869974 1.4420 0.1508
EFFRt−2 −0.0987549 0.0853303 −1.1573 0.2485
inflationt−1 0.642072 0.0674825 9.5146 0.0000
inflationt−2 0.266423 0.0698966 3.8117 0.0002
Mean dependent var 3.367600 S.D. dependent var 2.338486
Sum squared resid 203.6620 S.E. of regression 0.984794
R2
0.827580 Adjusted R2
0.822654
F(6, 210) 167.9932 P-value(F) 2.71e–77
ˆρ −0.027356 Durbin–Watson 2.040261
F-tests of zero restrictions
All lags of CUR F(2, 210) = 2.72763 [0.0677]
All lags of EFFR F(2, 210) = 1.28115 [0.2799]
All lags of inflation F(2, 210) = 226.081 [0.0000]
All vars, lag 2 F(3, 210) = 6.91477 [0.0002]
For the system as a whole —
Null hypothesis: the longest lag is 1
Alternative hypothesis: the longest lag is 2
Likelihood ratio test: χ2
9 = 152.117 [0.0000]
37

38. 8.6 Gretl output: Forecasting 2015 Forecast evaluation
statistics
Output1: forecasting for unemployment rate 2015
For 95% confidence intervals, t(210, 0.025) = 1.971
Forecasting 2015 CUR
CUR prediction std. error 95% interval
2015:1 5.566667 5.309577 0.238126 4.840155 - 5.779000
2015:2 5.400000 5.014595 0.452934 4.121715 - 5.907476
2015:3 5.166667 4.788693 0.645536 3.516131 - 6.061254
2015:4 5.000000 4.625052 0.806969 3.034254 - 6.215850
Forecast evaluation statistics
Mean Error 0.34885
Mean Squared Error 0.12452
Root Mean Squared Error 0.35287
Mean Absolute Error 0.34885
Mean Percentage Error 6.6425
Mean Absolute Percentage Error 6.6425
Theil’s U 1.983
Bias proportion, UM
0.97734
Regression proportion, UR
0.014591
Disturbance proportion, UD
0.008067
38

39. Output2: forecasting for effected federal funds rate 2015
For 95% confidence intervals, t(210, 0.025) = 1.971
Table 3: Forecasting 2015 EFFR
EFFR prediction std. error 95% interval
2015:1 0.110000 0.704119 0.829953 -0.931987 - 2.340225
2015:2 0.123333 1.105314 1.260607 -1.379752 - 3.590380
2015:3 0.136667 1.487447 1.603811 -1.674186 - 4.649080
2015:4 0.160000 1.810838 1.895295 -1.925403 - 5.547080
Forecast evaluation statistics
Mean Error −1.1444
Mean Squared Error 1.4668
Root Mean Squared Error 1.2111
Mean Absolute Error 1.1444
Mean Percentage Error −839.11
Mean Absolute Percentage Error 839.11
Theil’s U 78.886
Bias proportion, UM
0.89292
Regression proportion, UR
0.10707
Disturbance proportion, UD
8.8448e-006
39

40. Output3: effected forcasting for inflation rate 2015
For 95% confidence intervals, t(210, 0.025) = 1.971
Table 4: Forecasting 2015 inflation
EFFR prediction std. error 95% interval
2015:1 0.113661 0.892991 0.968780 -1.016789 - 2.802771
2015:2 2.095098 1.074964 1.175392 -1.242116 - 3.392043
2015:3 1.310701 1.355684 1.370750 -1.346509 - 4.057877
2015:4 0.780736 1.569104 1.526165 -1.439463 - 4.577670
Forecast evaluation statistics
Mean Error −0.14814
Mean Squared Error 0.56789
Root Mean Squared Error 0.75359
Mean Absolute Error 0.6582
Mean Percentage Error −185.35
Mean Absolute Percentage Error 209.69
Theil’s U 0.51574
Bias proportion, UM
0.038642
Regression proportion, UR
0.048866
Disturbance proportion, UD
0.91249
40

41. 8.7 Gretl outout: Vector Autoregression (for forecasting
2016)
VAR system, lag order 2
OLS estimates, observations 1960:4–2014:4 (T = 217)
Log-likelihood = −546.461
Determinant of covariance matrix = 0.0308961
AIC = 5.2301
BIC = 5.5571
HQC = 5.3622
Portmanteau test: LB(48) = 573.664, df = 414 [0.0000]
Equation 1: CUR
Coefficient Std. Error t-ratio p-value
const 0.122459 0.0713234 1.7170 0.0875
CURt−1 1.62800 0.0591725 27.5128 0.0000
CURt−2 −0.662925 0.0579556 −11.4385 0.0000
EFFRt−1 0.00854371 0.0213839 0.3995 0.6899
EFFRt−2 0.00436053 0.0209741 0.2079 0.8355
inflationt−1 0.0253736 0.0165872 1.5297 0.1276
inflationt−2 −0.0195600 0.0171805 −1.1385 0.2562
Mean dependent var 6.118280 S.D. dependent var 1.603038
Sum squared resid 12.30472 S.E. of regression 0.242062
R2
0.977832 Adjusted R2
0.977198
F(6, 210) 1543.838 P-value(F) 1.1e–170
ˆρ −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]
41

42. Equation 2: EFFR
Coefficient Std. Error t-ratio p-value
const 0.311688 0.248588 1.2538 0.2113
CURt−1 −0.950592 0.206237 −4.6092 0.0000
CURt−2 0.897253 0.201996 4.4419 0.0000
EFFRt−1 1.00956 0.0745306 13.5455 0.0000
EFFRt−2 −0.100519 0.0731023 −1.3750 0.1706
inflationt−1 0.0108904 0.0578122 0.1884 0.8508
inflationt−2 0.136338 0.0598803 2.2768 0.0238
Mean dependent var 5.373118 S.D. dependent var 3.616445
Sum squared resid 149.4742 S.E. of regression 0.843672
R2
0.947089 Adjusted R2
0.945577
F(6, 210) 626.4837 P-value(F) 4.8e–131
ˆρ 0.066791 Durbin–Watson 1.865396
F-tests of zero restrictions
All lags of CUR F(2, 210) = 10.7551 [0.0000]
All lags of EFFR F(2, 210) = 788.22 [0.0000]
All lags of inflation F(2, 210) = 8.18794 [0.0004]
All vars, lag 2 F(3, 210) = 12.1922 [0.0000]
42

43. Equation 3: inflation
Coefficient Std. Error t-ratio p-value
const 0.446412 0.290169 1.5385 0.1254
CURt−1 −0.541048 0.240735 −2.2475 0.0256
CURt−2 0.494421 0.235784 2.0969 0.0372
EFFRt−1 0.125446 0.0869974 1.4420 0.1508
EFFRt−2 −0.0987549 0.0853303 −1.1573 0.2485
inflationt−1 0.642072 0.0674825 9.5146 0.0000
inflationt−2 0.266423 0.0698966 3.8117 0.0002
Mean dependent var 3.367600 S.D. dependent var 2.338486
Sum squared resid 203.6620 S.E. of regression 0.984794
R2
0.827580 Adjusted R2
0.822654
F(6, 210) 167.9932 P-value(F) 2.71e–77
ˆρ −0.027356 Durbin–Watson 2.040261
F-tests of zero restrictions
All lags of CUR F(2, 210) = 2.72763 [0.0677]
All lags of EFFR F(2, 210) = 1.28115 [0.2799]
All lags of inflation F(2, 210) = 226.081 [0.0000]
All vars, lag 2 F(3, 210) = 6.91477 [0.0002]
For the system as a whole —
Null hypothesis: the longest lag is 1
Alternative hypothesis: the longest lag is 2
Likelihood ratio test: χ2
9 = 152.117 [0.0000]
43

44. 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

45. References
[1] The History of Recessions in the United States
http://useconomy.about.com/od/grossdomesticproduct/a/recession histo.htm
[2] A publication of the Board of Governors of the Federal Reserve System
http://www.federalreserve.gov/pf/pf.htm
45