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Introduction
At the start of most beginning economics courses we learn the
economics is a science aimed toward answering the following
questions:
1. What does society produce with its resources?
2. How does society perform this production?
3. Who receives the results?
We can thus look at history as many economic questions and
answers.
This search leads to another, perhaps more important question:
“Who is best suited to answer these questions?
Introduction
“Who is best suited to answer these questions?”
It has been argued that those who go into government honestly, and
altruistically, think that they should answer the questions for
others.
If this is the case, it follows that those in government will think it
best that they stay in government and that society would be
better off if government answered more of the economic
questions.
Conclusion: The Government will grow over time.
Introduction
Problem 1: To test this conclusion, we can look at the size of the
Government in the United States.
Size of Government: The number of economic questions answered
by the Government/total number of economic questions.
Extremes: Anarchy and Marxist Communism – Government does
not answer any economic questions. Soviet Communism and
social planning– Government answers all (or most) economic
questions.
All other societies lie between the two extremes. If we look at each
dollar of expenditure as a tool used in a modern economy
toward these answers, an apt measure of government size is:
Size of Government: G/GDP
Size of the Government: G/GDP
Government's Share of the Economy
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
1929 1935 1941 1947 1953 1959 1965 1971 1977 1983 1989 1995 2001
The above chart shows government expenditure as a
portion of GDP. Note that it is relatively flat.
Size of the Government: G/GDP
One issue: Why not include transfer payments?
Possible
problems:
Transfer payments
are ambiguous in this context because they are the
result of government deciding that one person is better off with
The chart on the previous slide was generated using data
someone else’s money. Still, the use of that money in the end is still
directly
from
the NIPA
Other sources show
decided by
a person,
and nottables.
the government.
government spending as a portion of GDP rising sharply. It
Transfers/GDP
is likely that the discrepancy
is caused by prices.
0.1
For example, government expenditures are valued at cost,
0.1 consumption expenditures are valued through the
while
market.
0.1
0.1
0.0
1947
1953
1959
1965
1971
1977
1983
1989
1995
2001
We include only those purchases made in the end by Government.
Modeling G/GDP: Trend
Government's Share of the Economy
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
1929 1935 1941 1947 1953 1959 1965 1971 1977 1983 1989 1995 2001
The red series includes WWII. It trends downward slightly. The
blue series excludes WWII and we can see that it trends upward.
Thus, using this series, and excluding WWII, we can see that
government does tend to grow over time.
Using other sources, the government clearly grows over time.
Modeling G/GDP: Cycles
Problem 2: In most economic texts, government spending is
considered exogenous. That is, that it is not predicted by the
particular model. Is it appropriate to consider the size of
government to be independent of any large force (such as
unemployment, etc.)?
To approach this question we need to look closely at the
characteristics of the series itself and its relationship to other
variables.
Modeling G/GDP: Cycles
For simplicity, we approached this task without WWII, creating
the following series.
Government's Share of the Economy
24%
22%
20%
18%
16%
14%
1947
1953
1959
1965
1971
1977
1983
1989
1995
2001
Modeling G/GDP: Cycles
Autocorrelation and partial-autocorrelation functions
Government's Share of the Economy: ACF and PACF
1.0
ACF
PACF
0.7
0.5
0.2
-0.1
-0.3
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Lag
Comments: Note that the series is highly correlated to itself at lag
1, indicating that it will be properly modeled with an AR(1)
process.
Modeling G/GDP: Cycles
Unit Root Test
ADF Test Statistic
-3.31 1% Critical Value*
5% Critical Value
10% Critical Value
-3.46
-2.87
-2.57
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(GOVSHARE)
Method: Least Squares
Variable
Coefficient
Std. Error t-Statistic Prob.
GOVSHARE(-1)
D(GOVSHARE(-1))
C
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
-0.04
0.48
0.01
0.01
0.06
0.00
0.27
0.26
0.00
0.00
956.05
2.11
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
-3.31 0.00
8.30 0.00
3.33 0.00
0.00
0.00
-8.55
-8.50
39.77
0.00
The Dickey Fuller Test tests whether or not the time series being examined is
stationary or evolutionary. The ADF statistic is not significant at the 99% level. Due
to this evidence that it is evolutionary, the series was pre-whitened by taking a
logarithmic transformation to remove the trend in variance, and then firstdifferenced to remove the trend in the mean.
Modeling G/GDP: Cycles
Government's Share of the Economy: Proportional Changes
11%
9%
7%
5%
3%
1%
-2%
-4%
-6%
-8%
1948
1954
1960
1966
1972
1978
1984
1990
1996
2002
The above chart shows the resulting series, which appears to have lost
most of its structure.
Modeling G/GDP: Cycles
Unit Root Test: Proportional Changes
ADF Test Statistic
-6.99 1% Critical Value*
5% Critical Value
10% Critical Value
-3.46
-2.87
-2.57
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(GOVSHARE)
Method: Least Squares
Variable
Coefficient
Std. Error t-Statistic Prob.
DLNGOVSHARE(-1)
D(DLNGOVSHARE(-1))
C
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
-0.47
-0.06
0.00
0.25
0.25
0.02
0.07
586.45
1.98
0.07
0.07
0.00
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
-6.99 0.00
-0.88 0.38
0.31 0.75
0.00
0.02
-5.26
-5.21
36.88
0.00
This Dickey-Fuller test gives evidence that the time series is now stationary as seen
by its ADF statistic being significant at the 99% level.
Modeling G/GDP: Cycles
Autocorrelation and partial-autocorrelation functions
Proportional Changes: ACF and PACF
0.6
0.5
0.4
ACF
PACF
0.3
0.2
0.1
-0.1
-0.2
-0.3
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Lag
Comments: Note that the series is still highly correlated to itself
at lag 1, and there is some correlation at lag 5, indicating an
ARMA 1,5 model.
Modeling G/GDP: Cycles
We can quickly reduce the residuals to white noise with an ARMA model.
Below is the actual and fitted graph for an AR(1), MA(5) model, followed by the
ACF and PACF of the residuals.
G/GDP: ARMA Model of Proportional Changes
11%
9%
7%
5%
3%
1%
-2%
-4%
-6%
-8%
1948
Actual
1954
1960
1966
1972
1978
1984
Fitted
1990
1996
2002
Modeling G/GDP: Cycles
G/GDP: Residuals for Proportional Changes ARMA Model - ACF and PACF
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
-0.2
-0.3
1
ACF
PACF
Q-Stat P Values (Right Scale)
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15
16 17 18 19 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3 4 3 5 3 6
Lag
Comments: We can see now that the residuals do not have any structure. The Pvalues and the Q-statistics are all well above the 5% level.
Modeling G/GDP with the Unemployment Rate
One possible explanation for movement in the government’s
share of the economy may be found in employment variables.
The basic intuition behind this is that the government will try to
stimulate the economy using fiscal policy. Thus government
spending will grow with unemployment while other GDP
component shrink.
To test this, we look to see if the unemployment rate can explain
a significant portion of the movement of government spending.
Modeling G/GDP with the Unemployment Rate
Unemployment Rate
Unemployment Rate
11
10
9
8
7
6
5
4
3
2
1948
1954
1960
1966
1972
1978
1984
1990
1996
2002
Modeling G/GDP with the Unemployment Rate
Granger Causality
Pairwise Granger Causality Tests
Lags: 2
Null Hypothesis:
UNRATE does not Granger Cause DLNGOVSHARE
DLNGOVSHARE does not Granger Cause UNRATE
Obs
222
F-Statistic
4.36271
0.1604
Probability
0.01388
0.8519
The Granger Causality test above gives some evidence that there is
a causal relationship from the unemployment rate (unrate) to the
proportional changes to the government’s share of the economy
(dlngovshare). This leads to the following model form:
DLNGOVSHARE = h(z)*DLNUNRATE + error
In other words, the fractional change in government share is some
function of lagged values of the fractional change in unemployment
rate, plus the usual error term. The unemployment rate will be
transformed to log differences for easy interpretation.
Modeling G/GDP with the Unemployment Rate
Unemployment Rate: Proportional Changes Form
Unemployment Rate: Proportional Changes
0.5
0.4
0.3
0.2
0.1
-0.1
-0.2
-0.3
1948
1954
1960
1966
1972
1978
1984
1990
1996
2002
Modeling G/GDP with the Unemployment Rate
Correlogram of dlnUnRate
Date: 06/01/04 Time: 14:16
Sample: 1948:1 2004:1
Included observations: 224
Autocorrelation
Partial Correlation
*|.
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AC
PAC
Q-Stat
Prob
1
-0.091
-0.091
1.8944
0.169
2
0.071
0.063
3.0455
0.218
3
-0.471
-0.465
53.864
0
4
-0.011
-0.105
53.89
0
5
0.012
0.062
53.925
0
6
0.01
-0.26
53.947
0
7
0.011
-0.085
53.974
0
8
0.003
0.055
53.976
0
9
-0.025
-0.19
54.129
0
10
-0.018
-0.104
54.209
0
Modeling G/GDP with the Unemployment Rate
ARMA Model for the Unemployment Rate in Proportional Changes
Dependent Variable: DLNUNRATE
Method: Least Squares
Sample(adjusted): 1949:1 2004:1
Included observations: 221 after adjusting endpoints
Variable
Coefficient Std. Error t-Statistic
Prob.
C
0.00
0.00
0.21
0.84
AR(3)
-0.7500
0.06
-13.47
0.00
MA(6)
-0.5674
0.07
-8.23
0.00
R-squared
0.35 Mean dependent var
0.00
Adjusted R-squared
0.34 S.D. dependent var
0.35
S.E. of regression
0.28 Akaike info criterion
0.33
Sum squared resid
17.60 Schwarz criterion
0.38
Log likelihood
-34.00 F-statistic
58.13
Durbin-Watson stat
2.26 Prob(F-statistic)
0.00
Inverted AR Roots
.45+.79i .45 -.79i
-0.91
Inverted MA Roots
0.91 .45 -.79i
.45+.79i -.45 -.79i
-.45+.79i
-0.91
Modeling G/GDP with the Unemployment Rate
Correlogram of the Residuals of the ARMA model
Date: 06/01/04 Time: 14:25
Sample: 1949:1 2004:1
Included observations: 221
Q-statistic probabilities adjusted for 2 ARMA term(s)
Autocorrelation
*|.
|
Partial Correlation
*|.
|
AC
PAC
Q-Stat
1
-0.132
-0.132
3.8966
Prob
.|.
|
.|.
|
2
0.019
0.002
3.9769
.|.
|
.|.
|
3
0.021
0.024
4.0764
0.043
4
-0.105
-0.1
6.5568
0.038
*|.
|
*|.
|
.|.
|
.|.
|
5
0.024
-0.003
6.6923
0.082
.|.
|
.|.
|
6
0.012
0.018
6.7264
0.151
7
-0.067
-0.062
7.7532
0.17
*|.
|
*|.
|
.|.
|
.|.
|
8
0.009
-0.019
7.7711
0.255
.|.
|
.|.
|
9
-0.024
-0.022
7.9023
0.341
.|.
|
*|.
10
-0.056
-0.06
8.6394
0.374
.|.
|
.|.
|
11
-0.013
-0.043
8.6817
0.467
.|.
|
.|.
|
12
-0.002
-0.008
8.6832
0.562
.|.
|
.|.
|
13
-0.017
-0.021
8.7522
0.645
.|.
|
.|.
|
14
-0.012
-0.034
8.7869
0.721
|
Modeling G/GDP with the Unemployment Rate
Derivation of the Distributed Lag Model w(t)
Using the derived AR(3) and MA(6) error structure from the DLNUNRATE time series, it is
possible to transform the original model (see “Proposed Model Form”) so that that the
DLNUNRATE term is approximately orthogonalized (Nun(t) is used to represent this new
term). Since all terms in the original equation must undergo the same transformation, a new
dependent variable is derived, which is referred to as w(t). In similar fashion, the transformed
error term is now referred to as residw(t). The exact procedure is as follows:
The Coefficients of the Zs (the lag operators) are the Betas from the ARMA(3,6) DLUNRATE
model.
w(t )  h( z ) * Nun (t )  resid w (t )
1  0.749958 * Z * d ln unrate
N (t ) 
1  0.56741* Z 
1  0.749958 * Z * d ln govshare
w(t ) 
1  0.56741* Z 
1  0.749958 * Z * resid
resid (t ) 
1  0.56741* Z 
3
un
6
3
6
3
w
6
(t )
govshare
w = (dlngovshare + .0749958*dlngovshare(-3)) / (dlngovshare0.567410*dlngovshare(-6))
Modeling G/GDP with the Unemployment Rate
Cross Correlation of W(t) and ResUnrate
Cross Correlations
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
-0.2
-0.3
-0.4
Lag
Lead
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Displacement
There is clearly significant correlations at lag 2 and lag 5.
Modeling G/GDP with the Unemployment Rate
Estimation of the distributed lag model
Dependent Variable: W
Method: Least Squares
Sample(adjusted): 1950:2 2004:1
Included observations: 216 after adjusting endpoints
Variable
Coefficient Std. Error t-Statistic
C
0.21
0.34
0.60
RESUNRATE(-2)
-9.5562
1.21
-7.90
RESUNRATE(-5)
19.1245
1.21
15.85
R-squared
0.59 Mean dependent var
Adjusted R-squared
0.59 S.D. dependent var
S.E. of regression
5.06 Akaike info criterion
Sum squared resid
5450.65 Schwarz criterion
Log likelihood
-655.14 F-statistic
Durbin-Watson stat
1.85 Prob(F-statistic)
Prob.
0.55
0.00
0.00
0.23
7.88
6.09
6.14
154.28
0.00
While this model performs well, it produces residuals with significant
structure as seen by the low p-values on the correlogram Q-statistics. The
ACF highlights lag 3 and 9 as candidates for AR processes. Estimation of
the above model with added AR(3) and AR(9) terms produces significant
coefficients and residuals without structure. They are not normal,
however.
Modeling G/GDP with the Unemployment Rate
Squared Residuals: Episodic Variance.
The residuals are not normal because the variance is non-constant, as seen
by the following chart of the squared residuals.
800
600
400
200
0
50
55
60
65
70
75
80
85
90
95
00
RESSQU_DISLAG
The problem can be solved using an ARCH model.
Modeling G/GDP with the Unemployment Rate
ARCH model:
Dependent Variable: W
Method: ML – ARCH
Sample(adjusted): 1952:3 2004:1
Included observations: 207 after adjusting endpoints
Convergence achieved after 100 iterations
Coefficient Std. Error z-Statistic
Prob.
C
0.93
0.05
18.12
0.00
RESUNRATE(-2)
-6.53
0.31
-20.87
0.00
RESUNRATE(-5)
11.50
0.57
20.21
0.00
AR(3)
0.10
0.03
3.17
0.00
AR(9)
0.07
0.02
3.56
0.00
Variance Equation
C
0.18
0.21
0.86
0.39
ARCH(1)
5.22
0.66
7.87
0.00
GARCH(1)
0.20
0.03
5.75
0.00
R-squared
0.53 Mean dependent var
0.18
Adjusted R-squared
0.51 S.D. dependent var
8.04
S.E. of regression
5.61 Akaike info criterion
5.54
Sum squared resid
6261.13 Schwarz criterion
5.67
Log likelihood
-565.66 F-statistic
32.09
Durbin-Watson stat
1.82 Prob(F-statistic)
0.00
Inverted AR Roots
0.77 .57+.46i
.57 -.46i .11 -.72i
.11+.72i
-.38+.66i -.38 -.66i -.68+.27i
-.68 -.27i
Modeling G/GDP with the Unemployment Rate
Forecasts
2 period forecast results
40
Forecast: WF
Actual: W
Forecast sample: 2004:1 2004:4
Adjusted sample: 2004:1 2004:3
20
Included observ ations: 1
0
-20
-40
2004:1
2004:2
WF
2004:3
± 2 S.E.
300
250
200
150
100
50
0
2004:1
2004:2
Forecast of Variance
2004:3
Root Mean Squared Error
6.517109
Mean Abs. Percent Error
Mean Absolute Percentage Error
6.517109
126.7705
Modeling G/GDP with the Unemployment Rate
Forecasts
Slight increase in government share.
40
20
0
-20
-40
02:1
02:3
03:1
W
FORECAST
03:3
04:1
04:3
05:1
FORECAST+2*SEF_WF
FORECAST-2*SEF_WF
Overall, the forecast is relatively “stable.”
Modeling G/GDP with the Unemployment Rate
Conclusions
Theoretically, government share should increase over time.
Forecasts predict a slight increase in government share over time.
Government share is not an exogenous variable (Econ 208).
Rather, it is influenced by other factors such as the unemployment
rate.