Assignment 2

Maastricht University

School of Business and Economics

Times-Series Methods and Dynamic Econometrics

Assignment II

Evidence of Okun law and Phillips Curve

Ali NabbiI6061881

Jonas StopschinskiI6053966

Giselle Carolina Fernandes SilvaI6115539

December 24, 2015

Contents

1 Introduction 2

2 Data and Graphical Analysis 32.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Graphical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Unit root tests 5

4 Structural Break and Unit root 7

5 Cointegration tests 8

6 Conclusion 11

1

Chapter 1

Introduction

This paper analyzes inflation in terms of a combination of variables such as, personal income,industrial production, employment, retail sales. In particular, the relations between inflation,unemployment rate, Gross Domestic Product(GDP) will be investigated. These authors ex-pect to observe evidences of Phillip’s curve or Okun law during this analysis. To investigatethe long-run relationship between the variables, different cointegration tests are performed. Ifa cointegrating vector can be found then long-run relationship between the variables can beformulated.

The structure of this paper can be summarized as follows. First, the series will be analyzedgraphically. Second, the order of integration of respected series will be determined using aug-mented Dickey-Fuller, Phillip-Perron method and KPSS test. Moreover, Zivot-Andrew’s test ischosen to test the order of integration with the possibility of structural break. Finally, once theorder of integration is determined, the cointegration relationships will be analyzed.

2

Chapter 2

Data and Graphical Analysis

2.1 Data

Data is provided from January 1959 to December 2003 for five time-series variables in US,namely inflation, personal income, industrial production, retail sales and employment. Theupdated dataset is downloaded from the Federal Reserve Bank research database. Additionally,the series unemployment rate and GDP are considered. Note that GDP is measured quarterlyand to adjust the time steps, monthly GDP is considered as linear interpolation of the quarterlymeasures.

The figure (2.1) illustrates inflation from January 1959 to December 1974 which seems tohave numerous missing values. In order to have a reliable data, the dataset from January 1975to August 2015 is considered. The time-series variables that are taken into this analysis arelisted below.

• Inflation

• GDP

• Unemployment rate

• Industrial Production

• Personal Income

Figure 2.1: Inflation from January 1959 to December 2003

3

CHAPTER 2. DATA AND GRAPHICAL ANALYSIS 4

2.2 Graphical Analysis

The figure (2.2) illustrates the levels and the first difference of GDP. It is self explanatory thatthe levels of GDP has a evident trend and the first difference of GDP seems to be somehowstationary. Similar conclusion goes the same with the other series. Note the levels and the firstdifference of the other series in the figures (A.1), (A.2) and (A.3).

Figure 2.2: The levels of GDP (left) and its first difference (right)

However, inflation solely on its levels seems to be stationary.

Figure 2.3: The levels of Inflation

For further analysis in the graphical approach, Autocorrelation Function(ACF) and PartialAutocorrelation Function(PACF) can show the first impression to analyze the order of integra-tion of the series.

The figures (A.4) - (A.13) illustrate the ACF and PACF of the five time-series both in thelevels and first difference. Almost all the ACF in all the series are decaying in their levels andthey have a cut-off on very first lags in PACF which could indicate the the order of integrationhigher than one. Note that this argument can not be entirely valid for the serie inflation.

Chapter 3

Unit root tests

To test for unit roots in the time series, and for that three test are chosen namely, the Argu-mented Dickey-Fuller test, Phillips-Perron test and KPSS test.

As seen before, from the graphical analysis, we suspect Inflation to be stationary and GDP,Industrial product, Personal Income and Unemployment rate to be either I(1) or I(2).

For the Argumented Dickey-Fuller test, the following hypothesis are tested:

H0 :ρ = p

vs.

H1 :ρ ≤ p

(3.1)

Following the Pantula principle, we first test for unit roots in the second order differencefor the time series and, if rejecting the null hypothesis, we will check for unit roots in the firstorder difference, then level data:

H0 :ρ = 2

vs.

H1 :ρ < 2

(3.2)

Then

H0 :ρ = 1

vs.

H1 :ρ < 1

(3.3)

Then

H0 :ρ = 0

vs.

H1 :ρ < 0

(3.4)

For the Phillips-Perron test, the following hypothesis are tested:

H0 :ρ > p

vs.

H1 :ρ = p

(3.5)

5

CHAPTER 3. UNIT ROOT TESTS 6

Finally, as a confirmation to the previous tests, the KPSS testis applied, which is conductedunder a null hypothesis of no unit root, against an alternative hypothesis of a unit root.

The results of the 3 tests, for each of the time series, on the first difference and for the leveldata are given in tables (3.1), (3.2) and (3.3).

PP ADF KPSS

Time Series p-value t-Stat p-value t-Stat Crit. Value 5% t-StatInflation < 0.01 -62.468 < 0.01 -14.117 0.146 0.008

GDP < 0.01 -22.544 < 0.01 -9.9807 0.146 0.023Personal Income < 0.01 -77.077 < 0.01 -15.003 0.146 0.007

Industrial Production < 0.01 -54.431 < 0.01 -11.018 0.146 0.018Unemployment rate < 0.01 -65.202 < 0.01 -11.145 0.146 0.011

Table 3.1: Unit Root and Stationarity tests for second difference data

PP ADF KPSS


GDP < 0.01 -7.111 < 0.01 -4.783 0.146 1.688Personal Income < 0.01 -29.333 < 0.01 -7.937 0.146 1.335

Industrial Production < 0.01 -18.182 < 0.01 -5.612 0.146 0.094Unemployment rate < 0.01 -20.313 < 0.01 -4.594 0.146 0.093

Table 3.2: Unit Root and Stationarity tests for first difference data

PP ADF KPSS


GDP 0.815 -1.441 0.632 -1.873 0.146 8.153Personal Income 0.848 -1.362 0.845 -1.371 0.146 8.113

Industrial Production 0.724 -1.655 0.387 -2.452 0.146 8.002Unemployment rate 0.674 -1.774 0.089 -3.192 0.146 1.108

Table 3.3: Unit Root and Stationarity tests for level data

Both ADF and PP tests indicate that the series Inflation is I(0) and GDP, Industrial product,Personal Income and Unemployment Rate are I(1).

KPSS indicates that Inflation is not, in fact, stationary in the level, suggesting that thisseries might present a trend in time. This test also suggests that GDP and Personal Incomeare not stationary in the first difference.

Despite having this contradiction in the results, the results obtained in the KPSS test mightbe related to a deterministic trend, instead of a stochastic trend, which would, then, excludethe presence of unit roots in Inflation, even on the presence of non stationarity by the KPSStest.

Chapter 4

Structural Break and Unit root

The downside of the classical unit root test such as Augmented Dickey-Fuller and Phillip-Perron’s test is that the is no consideration of the possible structural break in the data. Thefollowing tests are recommended in order to take structural breaks into unit root analysis:

• Lee and Strazicich’s unit root test

• Zivot and Andrew’s unit root test

Zivot and Andrews(ZA) unit root test allows a break at an unknown point in either theintercept, the linear trend or both. On the other hand, Lee and Strazicich’s unit root testapply multiple structural break to the intercept, deterministic trend or to both. In this paper,possibility of one break will be analyzed mainly because of the break caused by financial crisisin 2008. As a result, ZA unit root test is chosen.

In the levels both intercept and stationary trend are included in ZA test but in the firstdifference the trend can be excluded.

Series t-stat Break point

Inflation -13.6132 Sep. 1981∆Inflation -25.3529 May. 2015GDP -2.6115 Mar.1990∆GDP -7.7061 Feb. 2007Unemployment rate -3.7112 Apr. 2008∆Unemployment rate -19.9117 Oct. 2010Personal Income -5.4292 Dec. 1993∆Personal Income -28.1177 Apr. 2008Industrial Production -3.3263 Jun. 2008∆Industrial Production -7.7061 Feb. 2007

Critical value for levels is -5.08 and for difference -4.80

Table 4.1: ZA unit root test results

The null hypothesis of the ZA test is the existence of unit root in the respected series. Asa consequence, ZA test confirms the order of integration using ADF test or PP unit root testfrom the tables (3.1)-(3.3).

7

Chapter 5

Cointegration tests

The first cointegration test (Engle-Granger test) was done by applying the ADF-test on the errorterms of the regression of interest. For example to test if the variables GDP and unemploymentare cointegrated, a simple model of GDP on unemployment can be estimated by OLS. The ADFtest (as explained earlier) is then applied according to the Pantula principle to the error terms.If the error terms turn out to be I(0) then the variables are cointegrated, because there is a linearcombination of GDP and unemployment that is stationary (while the variables themselves areI(1); this was tested earlier). If the error terms are I(1), then the variables are not cointegrated,since no linear combination of GDP and unemployment could be found, that is stationary.

The second cointegration test is the Phillips-Ouliaris test. According to Phillips and Ouliaris,the unit root test on the residuals will not converge to the Dickey-Fuller distribution, becausethe test is affected by different aspects of the variables under consideration (e.g. the number oftrends and variables in the test). Consequently, new critical values are needed to interpret theresults, which come from the Phillips-Ouliaris distribution.

The third cointegration test applied to the variables is the Johansen-test. Here, the equationof interest is for example Yt = µ+ ζ × Yt−1 + ϵt. Subtracting Yt−1 from both sides leads to:

△Yt = µ+ (ζ − 1)× Yt−1 + ϵt (5.1)

Assuming, that Yt is I(1), the left side is clearly I(0). The error term ϵt and the constant arealso I(0) if there is cointegration, so the term (ζ − 1) × Yt−1 is I(0), too. Hence, the rank of(ζ − 1) provides information about how many different cointegrating relations there are. Inaddition to that, the matrix can be decomposed, such that (ζ − 1) = αβ′, where β contains thecointegrating vectors and α is used to adjust or scale β.

By applying the tests, as explained above, the following results were obtained (summarizedin table (5.1), (5) and (5.3)).

Table (5.1) tests the relations based on an ADF-test applied on the error terms. For exampleGDP on the unemployment rate (also keeping in mind that both are I(1) variables), but not onan intercept, then we find for the first difference of the resulting errors, a DF-statistic of -7.35.We reject the null of non-stationarity. Since the Dickey-Fuller statistic for the levels of theerrors leads to a test-value of -1.3, we do not reject the null of non-stationarity, indicating thatindeed, the error terms are I(1). The series are not cointegrated. Similar results can be found,by doing the same analysis with tests of cointegration between the other relations indicatedearlier (see table (5.1)). Obviously, this result is not entirely surprising. For example, accordingto Okun’s law, it is the first differences of GDP and unemployment, i.e. unempt − unempt−1

and GDPt − GDPt−1 that is cointegrated, not the levels. Redoing the test for this relation,leads to I(0) errors and thus is in accordance with the theory.

These relations have been tested with the Phillips-Ouliaris test as well (see table (5)). Thesame conclusion as for the ADF-based test was found here. The tests all conclude that the

8

CHAPTER 5. COINTEGRATION TESTS 9

Dickey-Fuller testCointegrated variables First Difference p-value Levels p-value

GDP,unemployment -7.3446 < 0.01 -1.2965 0.8761GDP, industrial prod -21.6880 < 0.01 -0.9132 0.9515GDP, personal income -31.7760 < 0.01 -1.8613 0.6370

GDP, retail sales -18.0490 < 0.01 -2.2475 0.4736unemp, industr. prod. -5.5272 < 0.01 -2.8132 0.2341unemp, pers income -5.3662 < 0.01 -2.6630 0.2977unemp, retail sales -5.4445 < 0.01 -2.2541 0.4708

indust prod, pers income -7.4341 < 0.01 -1.5835 0.7546indust prod, retail sales -6.3428 < 0.01 1.5828 0.9900pers income, retail sales -6.7075 < 0.01 1.3954 0.9900

Table 5.1: Cointegration tested with the Augmented Dickey-Fuller test on residuals

Phillips-Ouliaris testCointegrated variables test-statistic p-value

GDP,unemployment 1.8323 > 0.15GDP, industrial prod -6.3464 > 0.15GDP, personal income -10.4280 > 0.15

GDP, retail sales -15.4740 0.1362unemp, industr. prod. -7.7256 > 0.15unemp, pers income -7.7693 > 0.15unemp, retail sales -7.7611 > 0.15

indust prod, pers income -9.0274 > 0.15indust prod, retail sales 1.7166 > 0.15pers income, retail sales -0.7116 > 0.15

Table 5.2: Cointegration tested with the Phillips-Ouliaris test on the residuals

Johansen testCointegrated variables test-statistic

GDP,unemployment 3.20GDP, industrial prod 8.28GDP, personal income 14.73

GDP, retail sales 26.05unemp, industr. prod. 2.75unemp, pers income 13.94unemp, retail sales 18.43

indust prod, pers income 26.60indust prod, retail sales 18.63pers income, retail sales 22.74

Critical values: 90%: 12.91 95%: 14.90 99%: 19.19

Table 5.3: Cointegration tested with Johansen-test

CHAPTER 5. COINTEGRATION TESTS 10

relations are not cointegrated. Also, repeating the cointegration test for Okun’s law as in theprevious paragraph leads to the result of cointegration.

For the Johansen-test, the results are different. The tests between GDP and unemployment,industrial production, respectively and between unemployment and industrial production thetest statistics imply that there is no cointegration between these variables. However, for all theother tested relationships, the test-statistics suggest cointegration at least at the 90% confidenceinterval.

As before, the relationship of Okun’s law was tested as well and as before the test suggestscointegration of unempt − unempt−1 and GDPt −GDPt−1. The test value is 180.43 against acritical value of 19.19 at the 99% confidence interval.

Chapter 6

Conclusion

All tests of cointegration between the variables, lead to similar conclusions. According to thetheory, we could not expect to find cointegration between the levels of GDP and unemployment.However, we could confirm Okun’s law, since the first differences of GDP and unemployment arecointegrated according to all three tests of cointegration. Hence, the result is also consistent withthe theory. The tests for the other relations have to be investigated further, to really concludeanything about them. For the relationships of GDP and the other variables it would be worthinvestigating if the first differences are cointegrated, similar the GDP and unemployment inOkun’s law.

11

Appendix

Figure A.1: Unemployment in levels and first difference

Figure A.2: Industrial production in levels and first difference

Figure A.3: Personal income in levels and first difference

12

CHAPTER 6. CONCLUSION 13

Figure A.4: ACF and PACF of Inflation in levels

Figure A.5: ACF and PACF of Inflation in first difference


Figure A.6: ACF and PACF of GDP in levels

Figure A.7: ACF and PACF of GDP in first difference


Figure A.8: ACF and PACF of Unemployment rate in levels

Figure A.9: ACF and PACF of Unemployment rate in first difference


Figure A.10: ACF and PACF of Industrial Production in levels

Figure A.11: ACF and PACF of Industrial Production in first difference


Figure A.12: ACF and PACF of Personal Income in levels

Figure A.13: ACF and PACF of Personal Income in first difference

Assignment 2

Documents

Transcript of Assignment 2