Short run relationship between

the major economic indicatorsof

US economy(Project Report)

Course – Econometric Theory (MTH 676) Instructor – Dr. Sharmistha MitraBy,Aurkodyuti Das (Roll – Y8706)Sabyasachi Das (Roll – Y8715)M.Sc (2 Yr) StatisticsDepartment of Mathematics & StatisticsIndian Institute of Technology, Kanpur

IntroductionThe economic history of the United States has its roots

in European settlements in the 16th, 17th, and 18th centuries. The American colonies progressed from marginally successful

colonial economies to 13 small, independent farming economies, which joined together in 1776 to form the United States of America.

In 230 years the United States grew to a huge, integrated, industrialized economy that makes up over a quarter of the world economy.

Why U.S.?The economy of the United States is the largest national

economy in the world in both actual dollars and by Purchasing Power Parity.

Its nominal gross domestic product (GDP) was estimated as $14.4 trillion in 2008, which is about three times that of the world's second largest economy, Japan. The U.S. economy maintains a very high level of output per person (GDP per capita, $47,422 in 2008).

The U.S. economy has maintained a stable overall GDP growth rate, a low unemployment rate, and high levels of research and capital investment funded by both national and, because of decreasing saving rates, increasingly by foreign investors.

Why this topic?Early signals of recession or of recovery are of great interest to

business-people, policy makers, job seekers, and investors. Because such decision makers consider turning points in the aggregate level of economic activity to be of special importance, considerable effort has been spent to forecast when these turns will occur.

A reasonable way to forecast these turning points is to search for sectors of the economy that tend to lead the overall economy; observed turning points in these sectors would suggest that the overall economy will soon turn.

Objective of Study Our primary concern is to study the short run relationship

between the major economic indicators of US economy hence here we try to fit an Error Correction Model between the major economic indicators dependent for US economy. The ECM will describe the variation around its long term trend in terms of –

A set of I(0) exogenous variables. Variation of a set of variables that are co-integrated with the

regressand around their long term trend and error correction i.e. equilibrium error in the co-integration model.

Economic Indicators Variables Abbreviatio

nUnit

Real GDP gdp Billion of Dollars

Money Supply ms M2

Unemployment Rate unemp N/A

Inflation Rate inf N/A

Federal Fund Rate fed N/A

Exchange Rate exc N/A

Government Expenditure expnd Billion of Dollars

Net Export exp Billion of chained 2005 Dollars

Net Import imp Billion of chained 2005 Dollars

Crude Oil Price cop Dollars per Barrel

Government Receipt rcpt Billion of Dollars

Capacity Utilization cpu N/A

Data Sources U.S. Department of Commerce: Bureau of Economic Analysis http://research.stlouisfed.org/fred2/series/GDP?cid=106 U.S. Department of Labour: Bureau of Labour Statistics http://www.swivel.com/data_sets/show/1003612 Bureau of Labour Statistics

http://inflationdata.com/Inflation/Inflation_Rate/HistoricalInflation.aspx Illinois Oil & Gas Association http://www.ioga.com/Special/crudeoil_Hist.htm Board of Governors of the Federal Reserve System http://research.stlouisfed.org/fred2/series/FEDFUNDS?cid=118 Economic Time Series Page http://www.economagic.com Note: All are quarterly data & seasonally adjusted.Data Period: 1991(1st Quarter) to 2007 (1st Quarter)

Methodology, Analysis & ResultsNote: Software used SASExcept variables involving any type of rates we consider log of

the variables given below for interpreting their co-efficient more easily. There is another reason for taking the log transformation of those variables as sometime the log of the variables decrease the order of integration.

Real GDP, Money Supply, Government Expenditure, Crude oil Price, Government Receipt, Net Import, Net Export

Assume that all the variables are coming from different AR time series processes. First we determine the order of corresponding AR process of each variable (Using AIC(k) criterion) assuming maximum order to be AR(8).

Contd… Using AIC(k) criterion on all of the other variables in the same manner we

have the following result.Variable Order of AR process Value of min AIC(k)

Export Rate 2 -451.1787Import Rate 4 -503.0994

Capacity Utilization 2 -55.4091Real GDP 2 -552.5743

Government Receipt 8 -428.8282Government Expenditure 2 -528.3475

Exchange Rate 6 183.6709Federal Fund Rate 2 -143.556

Inflation Rate 5 -89.6426Money Supply 6 -591.4822

Crude Oil Price 6 -239.405Unemployment Rate 3 -204.4055

Contd…Now we plot each of the above series to know whether they have

a zero mean, some non zero mean or some trend with in them.

Then we determine the order of the integration of each variable listed above (Using Augmented Dicky-Fuller test).

Hypothesis of testing of ADF Test:

H0: the errors are non stationary v/s H1: the errors are stationary.

(whenever we check for stationarity of a series, these will be always the corresponding hypotheses ).

Contd…Graph (for exp)

From the above graph we see that there is a linear trend.

exp

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7.1

7.2

7.3

7.4

t

0 10 20 30 40 50 60 70

Contd…SAS Output for ADF (for exp)

Augmented Dickey-Fuller Unit Root Tests

Type Lags Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean 0 0.1274 0.7090 5.61 0.9999 2 0.1173 0.7064 2.47 0.9965Single Mean 0 -0.8557 0.8960 -1.30 0.6260 17.13 0.0010 2 -0.9430 0.8881 -0.85 0.7967 3.51 0.1877Trend 0 -3.3824 0.9152 -1.52 0.8140 1.55 0.8670 2 -7.5175 0.5990 -1.89 0.6480 1.84 0.8108

Now 0.648 > 0.01 so we accept the null hypothesis & conclude that at 1% level of significance exp series is non stationary. So we check stationarity of ∆(exp) series.

Contd…Graph (for ∆(exp))

From the above graph we see that there is a some non zero mean.

exp1

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

t1

0 10 20 30 40 50 60 70

Contd…SAS Output for ADF (for ∆(exp))

Augmented Dickey-Fuller Unit Root Tests Type Lags Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean 0 -25.7335 0.0001 -4.09 <.0001 2 -16.5325 0.0032 -2.68 0.0080Single Mean 0 -37.8805 0.0006 -5.18 0.0001 13.43 0.0010 2 -38.6198 0.0006 -3.73 0.0058 6.95 0.0010Trend 0 -38.1604 0.0003 -5.15 0.0004 13.28 0.0010 2 -39.4744 0.0002 -3.70 0.0297 6.88 0.0398 Now 0.0058 < 0.01 so we reject the null hypothesis & conclude that at 1%

level of significance ∆(exp) series is stationary. So, Net Export is I(1) variable.

Contd…Now we compute tha same analysis in the same manner for all

the variables & From the above results we group all the variables in three groups namely I(0), I(1), I(2).

Here is the grouping of the variables.

Now we take log(gdp) as regressand. log (gdp) is I(1) series.Now we regress log(gdp) on the other I(1) variables and obtain

the residuals and check whether these variables are cointegrated are not.

Order of Integration VariablesI(0) exc

I(1) expnd, gdp, cpu, exp, fed, inf, cop, unemp

I(2) rcpt, imp, ms

Contd…SAS Output of Model 

Parameter Estimates 

Parameter Standard Variable Label DF Estimate Error t Value Pr > |t|

  expnd expnd 1 -0.00506 0.00336 -1.51 0.1373

cpu cpu 1 0.01293 0.00004919 262.92 <.0001 exp exp 1 0.00621 0.00428 1.45 0.1524

fed fed 1 -0.00004506 0.00017194 -0.26 0.7942 inf inf 1 0.00061217 0.00023620 2.59 0.0121

cop cop 1 0.00077109 0.00085790 0.90 0.3725 unemp unemp 1 0.00062050 0.00058642 1.06 0.2945

Now we assume that the residuals are coming from AR process. Using AIC(k) criterion we obtain the order of the corresponding AR process to be 5.

Now we plot the residuals to know whether they have a zero mean, some non zero mean or some trend with in them.

Contd…

From the above graph we see that there is a zero mean.

Now we apply ADF test on this residuals to check whether they are stationary or not.

Residual

-0.0020

-0.0018

-0.0016

-0.0014

-0.0012

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

t

0 10 20 30 40 50 60 70

Contd…SAS Output for ADF

Augmented Dickey-Fuller Unit Root Tests Type Lags Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean 0 -11.7285 0.0146 -2.46 0.0146 5 -36.0968 <.0001 -2.67 0.0084Single Mean 0 -11.7155 0.0757 -2.43 0.1366 2.98 0.3204 5 -36.3670 0.0005 -2.64 0.0900 3.50 0.1903Trend 0 -11.6607 0.2982 -2.40 0.3738 2.98 0.5867 5 -35.2510 0.0006 -2.63 0.2710 3.57 0.4716 Now .0084 < .0001 so we reject the null hypothesis & conclude that at 1%

level of significance residuals are stationary.

Contd… So the co integration vector between log(gdp), log(expnd), cpu,

log(exp), fed, inf, log(cop), unemp is(1 0.00506 -0.01293 -0.00621 0.00004506 -0.00061217 -0.00077109 -0.0006205).

The variables which are I(2) series we take 2nd order differencing on them and make it I(0) .We also take 1st order differencing of all the variables belonging to I(1) series to make them I(0).

Now we obtain our ECM model as,

∆log(gdpt) = c1*exct + c2*∆2log(rcptt) + c3*∆2log(mst) + c4*∆2log(impt) + c5*∆log(expndt) + c6*∆log(cput) + c7*∆log(expt) + c8*∆(fedt) + c9*∆(inft) + c10*∆log(copt) + c11*∆(unempt) + c12*(ret-

1) + ut

Contd…SAS Output for Model

Parameter Estimates 

Parameter Standard Variable Label DF Estimate Error t Value Pr > |t|  exc exc 1 -8.81708E-7 8.888744E-7 -0.99 0.3259 rcpt2 rcpt2 1 -0.00008969 0.00152 -0.06 0.9531 imp2 imp2 1 -0.00227 0.00408 -0.56 0.5794 ms2 ms2 1 -0.00565 0.01013 -0.56 0.5791 expnd1 expnd1 1 -0.00203 0.00567 -0.36 0.7216 cpu1 cpu1 1 0.01272 0.00010309 123.37 <.0001 exp1 exp1 1 0.00908 0.00372 2.44 0.0182 fed1 fed1 1 -0.00002423 0.00019205 -0.13 0.9001 inf1 inf1 1 0.00011427 0.00013380 0.85 0.3971 cop1 cop1 1 0.00091805 0.00056531 1.62 0.1105 unemp1 unemp1 1 -0.00010274 0.00043777 -0.23 0.8154 re1 re1 1 -0.10793 0.07200 -1.50 0.1400

Contd… Now we assume that the residuals are coming from AR process. Using AIC(k)

criterion we obtain the order of the corresponding AR process to be 1.  Now we plot residuals of the above model.

From the above graph we see that there is a zero mean.

Residual

-0.0010

-0.0009

-0.0008

-0.0007

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

t

0 10 20 30 40 50 60 70

Contd… Now we apply ADF test on this residuals to check whether they are

stationary or not.SAS Output of ADF

Augmented Dickey-Fuller Unit Root Tests Type Lags Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean 0 -27.6449 <.0001 -4.17 <.0001 1 -36.1794 <.0001 -4.17 <.0001Single Mean 0 -27.6680 0.0006 -4.14 0.0017 8.55 0.0010 1 -36.2503 0.0006 -4.14 0.0017 8.56 0.0010Trend 0 -28.2935 0.0053 -4.18 0.0082 8.77 0.0010 1 -37.6470 0.0003 -4.21 0.0077 8.89 0.0010 Now we reject the null hypothesis & conclude that at 1% level of significance

residuals are stationary.

Contd…Check for Multicollinearity: We calculate VIF for each regressors except the Error Correction term in

the ECM model and then check it.SAS Output for Multicollinearity checking

Parameter Estimates  Parameter Standard VarianceVariable Label DF Estimate Error t Value Pr > |t| Inflation

Intercept Intercept 1 -0.00191 0.00070106 -2.72 0.0089 0exc exc 1 0.00001523 0.00000598 2.55 0.0140 1.54252rcpt2 rcpt2 1 0.00036877 0.00144 0.26 0.7990 1.07989imp2 imp2 1 -0.00293 0.00385 -0.76 0.4496 1.17432ms2 ms2 1 -0.00113 0.00969 -0.12 0.9079 1.30405expnd1 expnd1 1 -0.00107 0.00535 -0.20 0.8430 1.17672cpu1 cpu1 1 0.01266 0.00009916 127.71 <.0001 2.10644exp1 exp1 1 0.01113 0.00359 3.10 0.0032 1.80972fed1 fed1 1 0.00011189 0.00018780 0.60 0.5540 2.59207inf1 inf1 1 0.00013187 0.00012628 1.04 0.3014 1.49912cop1 cop1 1 0.00111 0.00053746 2.06 0.0442 1.61606unemp1 unemp1 1 -0.00024238 0.00041580 -0.58 0.5626 2.39781re1 re1 1 -0.14078 0.06893 -2.04 0.0464 1.21371

Contd…we see that VIF for all the regressors are < 5, so there is no

multicollinearity present in the model.

Check for Heteroscedasticity of €t’s:

We perform Breusch – Pagan – Godfrey Test and White’s Test.

Hypothesis of Testing of Heteroscedasticity:

H0: the errors are homoscedastic v/s H1: the errors are heteroscedastic.

Contd…SAS Outputs for Breusch - Pagan & White’s Test

Heteroscedasticity Test Equation Test Statistic DF Pr > ChiSq Variables  gdp1 White's Test 63.00 62 0.4407 Cross of all vars Breusch-Pagan 14.94 12 0.2446 1, exc, rcpt2, imp2, ms2, expnd1, cpu1, exp1, fed1, inf1, cop1, unemp1, re1 Now obsd white’s test statistic = 63 which is less than from both values.

[88.379 (tab value of chi square at (0.01,60)), 100.425(tab value of chi square at (0.01,70))]& obsd Breusch Pagan test statistic = 14.94 which is less than from both values [26.217(tab value of chi square at (0.01,12))], so we accept the null hypothesis at 1% level of significance & conclude that the residuals are homoscedastic.

Contd…Check for Autocorrelation of €t’s:

From the above graph we do not observe any type of auto correlation between the residuals. So OLS estimation is justified.

Residual

-0.0010

-0.0009

-0.0008

-0.0007

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

t

0 10 20 30 40 50 60 70

Conclusion So our final ECM model is

∆(log(gdpt)) = -8.81708E-7 * (exct) - 0.00008969 * ∆2(log(rcptt)) - 0.00227 * ∆2(log(impt)) - 0.00565 * ∆2(log(mst)) - 0.00203 * ∆(log(expndt)) + 0.01272 * ∆(cput) + 0.00908 * ∆(log(expt)) – 0.00002423 * ∆(fedt) + 0.00011427 * ∆(inft) + 0.00091805 * ∆(log(copt)) - 0.00010274 * ∆(unempt) -0.10793 * ret-1

Now we see that for the above model R square and adjusted R square are 0.9986 & 0.9983 respectively which is very good.

Now for the above model the error correction term is

ret-1= log(gdpt-1) +0.00506 * log(expndt-1) - 0.01293 * cput-1 - 0.00621 * log(expt-1) + 0.00004506 * fedt-1 - 0.00061217 * inft-1 - 0.00077109 * log(copt-1) - 0.0006205 * unempt-1

Contd… Now log(gdpt-1) = -0.00506 * log(expndt-1) + 0.01293 * cput-1 + 0.00621 *

log(expt-1)- 0.00004506 * fedt-1 + 0.00061217 * inft-1 + 0.00077109 * log(copt-1) + 0.0006205 * unempt-1

is the long run relationship among these variables. The ECM model says that change in log(gdp) in period t equals to sum of changes in all other variables (after proper differencing) in period t plus a correction for the discrepancy between log(gdp) and its equilibrium value in period t-1. If the model is in equilibrium in the period t-1 , the error correction term plays no role in change in log(gdp)in period t. If log(gdp) is below its equilibrium value, it needs to increase for the model to converge towards equilibrium which is assured here as coefficient of error correction term is -0.10793 which is negative and similarly when log(gdp) is above its equilibrium value, the negative value ensures that log(gdp) declines towards equilibrium when all other variables remain constant.

Questions?

Thank You