IntroductionAt 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.

IntroductionProblem 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

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

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/GDPPossible problems:

The chart on the previous slide was generated using data directly from the NIPA tables. Other sources show government spending as a portion of GDP rising sharply. It is likely that the discrepancy is caused by prices.

For example, government expenditures are valued at cost, while consumption expenditures are valued through the market.

Size of the Government: G/GDPOne issue: Why not include transfer payments?Transfer payments are ambiguous in this context because they are the result of government deciding that one person is better off with someone else’s money. Still, the use of that money in the end is still decided by a person, and not the government.

Transfers/GDP

0.0

0.1

0.1

0.1

0.1

1947 1953 1959 1965 1971 1977 1983 1989 1995 2001

We include only those purchases made in the end by Government.

Modeling G/GDP: TrendGovernment's Share of the Economy

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

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: CyclesProblem 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: CyclesFor simplicity, we approached this task without WWII, creating the following series.

Government's Share of the Economy

14%

16%

18%

20%

22%

24%

1947 1953 1959 1965 1971 1977 1983 1989 1995 2001

Modeling G/GDP: CyclesAutocorrelation and partial-autocorrelation functions

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.

Government's Share of the Economy: ACF and PACF

-0.3

-0.1

0.2

0.5

0.7

1.0

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

Lag

ACF

PACF

Modeling G/GDP: CyclesUnit Root Test

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 first-differenced to remove the trend in the mean.

ADF Test Statistic -3.31 1% Critical Value* -3.465% Critical Value -2.8710% Critical Value -2.57

Variable Coefficient Std. Error t-Statistic Prob.

GOVSHARE(-1) -0.04 0.01 -3.31 0.00D(GOVSHARE(-1)) 0.48 0.06 8.30 0.00C 0.01 0.00 3.33 0.00

R-squared 0.27 Mean dependent var 0.00Adjusted R-squared 0.26 S.D. dependent var 0.00S.E. of regression 0.00 Akaike info criterion -8.55Sum squared resid 0.00 Schwarz criterion -8.50Log likelihood 956.05 F-statistic 39.77Durbin-Watson stat 2.11 Prob(F-statistic) 0.00

*MacKinnon critical values for rejection of hypothesis of a unit root.Augmented Dickey-Fuller Test EquationDependent Variable: D(GOVSHARE)Method: Least Squares

Modeling G/GDP: CyclesGovernment's Share of the Economy: Proportional Changes

-8%

-6%

-4%

-2%

1%

3%

5%

7%

9%

11%

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: CyclesUnit Root Test: Proportional Changes

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.

ADF Test Statistic -6.99 1% Critical Value* -3.465% Critical Value -2.8710% Critical Value -2.57

Variable Coefficient Std. Error t-Statistic Prob.

DLNGOVSHARE(-1) -0.47 0.07 -6.99 0.00D(DLNGOVSHARE(-1)) -0.06 0.07 -0.88 0.38C 0.00 0.00 0.31 0.75

R-squared 0.25 Mean dependent var 0.00Adjusted R-squared 0.25 S.D. dependent var 0.02S.E. of regression 0.02 Akaike info criterion -5.26Sum squared resid 0.07 Schwarz criterion -5.21Log likelihood 586.45 F-statistic 36.88Durbin-Watson stat 1.98 Prob(F-statistic) 0.00

Method: Least Squares

*MacKinnon critical values for rejection of hypothesis of a unit root.Augmented Dickey-Fuller Test EquationDependent Variable: D(GOVSHARE)

Modeling G/GDP: CyclesAutocorrelation and partial-autocorrelation functions

Proportional Changes: ACF and PACF

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

0.6

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

ACF PACF

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: CyclesWe 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

-8%-6%-4%-2%1%3%5%7%9%

11%

1948 1954 1960 1966 1972 1978 1984 1990 1996 2002

Actual Fitted

Modeling G/GDP: Cycles

Comments: We can see now that the residuals do not have any structure. The P-values and the Q-statistics are all well above the 5% level.

G/GDP: Residuals for Proportional Changes ARMA Model - ACF and PACF

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

0.6

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

0

0.2

0.4

0.6

0.8

1

ACF PACF Q-Stat P Values (Right Scale)

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

2

3

4

5

6

7

8

9

10

11

1948 1954 1960 1966 1972 1978 1984 1990 1996 2002

Modeling G/GDP with the Unemployment Rate

Granger Causality

Null Hypothesis: Obs F-Statistic ProbabilityUNRATE does not Granger Cause DLNGOVSHARE 222 4.36271 0.01388

0.1604 0.8519DLNGOVSHARE does not Granger Cause UNRATE

Pairwise Granger Causality TestsLags: 2

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

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

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   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

Variable Coefficient Std. Error t-Statistic Prob.

C 0.00 0.00 0.21 0.84AR(3) -0.7500 0.06 -13.47 0.00MA(6) -0.5674 0.07 -8.23 0.00

R-squared 0.35 0.00Adjusted R-squared 0.34 0.35S.E. of regression 0.28 0.33Sum squared resid 17.60 0.38Log likelihood -34.00 58.13Durbin-Watson stat 2.26 0.00Inverted AR Roots .45+.79i .45 -.79iInverted MA Roots 0.91 .45 -.79i .45+.79i -.45 -.79i

-.45+.79i

Mean dependent var

Dependent Variable: DLNUNRATEMethod: Least SquaresSample(adjusted): 1949:1 2004:1Included observations: 221 after adjusting endpoints

Prob(F-statistic)-0.91

-0.91

S.D. dependent varAkaike info criterionSchwarz criterionF-statistic

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 Prob

*|. | *|. | 1 -0.132 -0.132 3.8966  

.|. | .|. | 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 RateDerivation 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.

unrated

Z

ZtNun ln*

*56741.01

*749958.01)(

6

3

)()(*)()( tresidtNzhtw wun

govshared

Z

Ztw ln*

*56741.01

*749958.01)(

6

3

)(*

*56741.01

*749958.01)(

6

3

tresidZ

Ztresid govsharew

w = (dlngovshare + .0749958*dlngovshare(-3)) / (dlngovshare-0.567410*dlngovshare(-6))

Modeling G/GDP with the Unemployment RateCross Correlation of W(t) and ResUnrate

Cross Correlations

-0.4-0.3-0.2-0.10.10.20.30.40.50.60.7

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

Lag Lead

There is clearly significant correlations at lag 2 and lag 5.

Modeling G/GDP with the Unemployment RateEstimation of the distributed lag model

Variable Coefficient Std. Error t-Statistic Prob. C 0.21 0.34 0.60 0.55

RESUNRATE(-2) -9.5562 1.21 -7.90 0.00RESUNRATE(-5) 19.1245 1.21 15.85 0.00

R-squared 0.59 0.23Adjusted R-squared 0.59 7.88S.E. of regression 5.06 6.09Sum squared resid 5450.65 6.14Log likelihood -655.14 154.28Durbin-Watson stat 1.85 0.00

Method: Least SquaresDependent Variable: W

Sample(adjusted): 1950:2 2004:1Included observations: 216 after adjusting endpoints

Mean dependent varS.D. dependent varAkaike info criterionSchwarz criterionF-statisticProb(F-statistic)

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 RateSquared Residuals: Episodic Variance.

The residuals are not normal because the variance is non-constant, as seen by the following chart of the squared residuals.

0

200

400

600

800

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 RateARCH model:

Coefficient Std. Error z-Statistic Prob. C 0.93 0.05 18.12 0.00

RESUNRATE(-2) -6.53 0.31 -20.87 0.00RESUNRATE(-5) 11.50 0.57 20.21 0.00

AR(3) 0.10 0.03 3.17 0.00AR(9) 0.07 0.02 3.56 0.00

C 0.18 0.21 0.86 0.39ARCH(1) 5.22 0.66 7.87 0.00

GARCH(1) 0.20 0.03 5.75 0.00R-squared 0.53 0.18Adjusted R-squared 0.51 8.04S.E. of regression 5.61 5.54Sum squared resid 6261.13 5.67Log likelihood -565.66 32.09Durbin-Watson stat 1.82 0.00Inverted AR Roots 0.77 .57+.46i .57 -.46i .11 -.72i

.11+.72i -.38+.66i -.38 -.66i -.68+.27i

Mean dependent varS.D. dependent varAkaike info criterionSchwarz criterionF-statisticProb(F-statistic)

-.68 -.27i

Dependent Variable: WMethod: ML – ARCHSample(adjusted): 1952:3 2004:1Included observations: 207 after adjusting endpointsConvergence achieved after 100 iterations

Variance Equation

Modeling G/GDP with the Unemployment RateForecasts

-40

-20

0

20

40

2004:1 2004:2 2004:3

WF ± 2 S.E.

Forecast: WFActual: WForecast sample: 2004:1 2004:4Adjusted sample: 2004:1 2004:3

Included observations: 1

Root Mean Squared Error 6.517109

Mean Abs. Percent Error 6.517109Mean Absolute Percentage Error 126.7705

0

50

100

150

200

250

300

2004:1 2004:2 2004:3

Forecast of Variance

2 period forecast results

Modeling G/GDP with the Unemployment RateForecasts

-40

-20

0

20

40

02:1 02:3 03:1 03:3 04:1 04:3 05:1

WFORECAST

FORECAST+2*SEF_WFFORECAST-2*SEF_WF

Slight increase in government share.

Overall, the forecast is relatively “stable.”

Modeling G/GDP with the Unemployment RateConclusions

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 unemploymentrate.