Macroeconometrics - handout 2

Macroeconometrics - handout 2

Piotr Wojcik, Katarzyna [email protected], [email protected]

March 14, 2007

1 Introduction

1.1 Absolute Income Hypothesis

English economist John Maynard Keynes (1883-1946) proposed the Absolute Income Hypoth-esis (J. M. Keynes, The General Theory of Employment, Interest and Money, New York, 1936) aspart of his work on the relationship between income and consumption. He stated that consump-tion is a function of income. If income rises, consumption will also rise but not necessarily at thesame rate. Absolute income hypothesis was much refined during the 1960s and 1970s, notably byAmerican economist James Tobin (1918-2002). In its developed form, absolute income hypothesisis still generally accepted.

CONSt = α + βINCt + εt (1)

where consumption and income are measured in real terms per capita. According to AIH themarginal propensity to consume (MPC) represented in the above equation by parameter β shouldbe greater than zero but less than one. As far as the constant term α is concerned - it should bepositive.

Lets check it empirically on dhsy.wf1 dataset.First we will check if there is any correlation between consumption and income.

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Piotr Wojcik, Katarzyna Rosiak-Lada Macroeconometrics 2007, handout2

Definitely there is strong correlation between consumption and income.

Task: Calculate Pearson correlation coefficient between consumption and income.

As You can see correlation is about 0.999, which is extremely high.

Task: Run a regression of consumption on income and verify Absolute Income Hypothesis.

Variable Coefficient Std. Error t-Statistic Prob.inc 0.859174 0.003458 248.4531 0.0000c -4527.759 947.5534 -4.778368 0.0000

Estimate of marginal propensity to consume seems to be in line with Absolute Income Hypoth-esis, but the constant is negative.

1.2 Permanent Income Hypothesis

About 20 years later an American economist Milton Friedman (1912-1992) developed the Per-manent Income Hypothesis (M. Friedman, A Theory of the Consumption Function, NationalBureau of Economic Research, Princeton, 1957). In its simplest form, permanent income hypoth-esis states that the choices consumers make regarding their consumption patterns are determinednot only by current income but by their longer-term income expectations. Friedman argues thatit would be more sensible for people to use current income, but also at the same time to formexpectations about future levels of income and the relative amounts of risk. Measured income andmeasured consumption contain a permanent (anticipated and planned, so permanent income =past income + expected future income) element and a transitory element (income that is earnedin excess of, or perceived as an unexpected windfall). Friedman concluded that the individual willconsume a constant proportion of his/her permanent income. Other conclusions are that low in-come earners have a higher propensity to consume and high income earners have a higher transitoryelement to their income and a lower than average propensity to consume.

CONSt = β1CONSt−1 + β2INCt + εt (2)

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where β2 is the short run marginal propensity to consume. Long run propensity to consumeis derived by solving above equation for long run consumption (assuming CONS∗ = CONSt =CONSt−1) and is equal β2

1−β1. It can be easily calculated a command window using command

scalar pih=c(2)/(1-c(1)). Lets verify the hypothesis.

Task: Run a regression of consumption on lagged consumption and income and check Perma-nent Income Hypothesis (are parameters at income and lagged consumption positive?).

1.3 Life-cycle hypothesis

Another interesting hypothesis is a life-cycle hypothesis (A. Ando and F. Modigliani, Testsof the Life Cycle Hypothesis of Saving: Comments and Suggestions, Oxford Institute of StatisticsBulletin, vol. XIX, May, 1957, pp. 99-124). Life-cycle hypothesis assumes that individuals consumea constant percentage of the present value of their life income. This is dictated by preferences andtastes and income. Ando and Modigliani argued that the average propensity to consume is higher inyoung and old households, whose members are either borrowing against future income or runningdown life-savings. Middle-aged people tend to have higher incomes with lower propensities toconsume and higher propensities to save.

2 DHSY model estimation - ADL models and general-to-specific

The paper of Davidson, J. E. H., Hendry, D. F., Srba, F. and Yeo, S., 1978, Econometric modellingof the aggregate time-series relationship between consumers expenditure and income in UnitedKingdom, Economic Journal 88, pp. 661-692, DHSY in short, has been very influential article onmodelling the consumption-income relationship, and it has inspired a large number of follow-upstudies. It has contributed to the popularity of the concept of error correction, introduced byPhillips (1957), for which statistical justification arrived later in the form of cointegration (see forexample Granger, C. W. J., 1981, Some properties of time series data and their use in econometricmodel specification, Journal of Econometrics 16, pp. 121-130).

Some of the data definitions have been changed, so that the old and the revised data are notcompatible.

We will begin by estimating the DHSY equation augmented by the intercept. It is based onfour- quarter differences of the logarithmic series and has the following form:

∆4const = β1∆4inct + β2∆∆4inct + β3∆4DOt + β4(const−4 − inct−4) + β5inft + β6∆inft + εt (3)

where cons is the natural logarithm of aggregate consumption, inc is the natural logarithm ofincome, the yearly rate of inflation is approximated by inft = ∆4pt and where pt is the naturallogarithm of the price deflator and DOt is a dummy variable related to periods when budgetannouncements might have caused special effects.

We will try to determine whether the analogous function can be derived from a more generalAutoregressive Distributed Lags (ADL) model.

In ADL model a dependent variable yt is expressed as a function of its own lagged valuestogether with current and lagged values of other independent variables.

In general-to-specific modelling first the kind of most general model is formulated. Then bytesting subsequent restrictions the general model is reduced and gets a final specific form.

In the original model the maximum lag length:

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• for consumption was equal to 4

• for income was equal to 5

• for inflation was equal to 1

The fifth lag for income is not directly obvious, but note that:

∆∆4inct = ∆(inct − inct−4) = (inct − inct−4) − (inct−1 − inct−5) (4)

It seems reasonable to start from a general ADL model with 5 lags for all variables (might be lowerfor inflation), so it will be ADL(5,5,5) model.

const = α0 + α1const−1 + α2const−2 + α3const−3 + α4const−4 + α5const−5 +β0inct + β1inct−1 + β2inct−2 + β3inct−3 + β4inct−4 + β5inct−5 +γ0inft + γ1inft−1 + γ2inft−2 + γ3inft−3 + γ4inft−4 + γ5inft−5

First lets calculate desired variables using command window:genr lcons=log(cons)genr linc=log(inc)genr inf=p-p(-4)

Now we will estimate the general model. Choose variables and open them as a group and writean equation.

lcons lcons(-1 to -5) linc linc(-1 to -5) infl infl(-1 to -5) c

As in the DHSY model inflation appears with maximum first lag, therefore the first step of modelreduction will be testing if lags of inflation higher than one are jointly statistically significant. Theestimation results show that none of the inflation lags is individually significant, but we have torun a joint test. In View/Representations one can see that coeficients at these lags are C(14)-C(17).

Lets go to View/Coefficients tests/Wald - Coefficient Restrictions... and write the null hypoth-esis there:

c(14)=0, c(15)=0, c(16)=0, c(17)=0.

The test F-statistic equals 1.088289 (p − value = 0.3681), therefore the null cannot be rejected, sothe model can be reduced. The reduced version will take a form:

const = α0 + α1const−1 + α2const−2 + α3const−3 + α4const−4 + α5const−5 +β0inct + β1inct−1 + β2inct−2 + β3inct−3 + β4inct−4 + β5inct−5 +γ0inft + γ1inft−1

Lets estimate the reduced model. In the results one can observe that second and third lags of con-sumption and income are not significant and this will be tested next - in the DHSY model secondand third lag do not appear.

Task: Check coefficient numbers and verify the null hypothesis about nonsignificance of second

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and third lag for consumption and income.

The test F-statistic equals 0.499669 (p − value = 0.7360), so again the null cannot be rejectedand our model can be further reduced to a form:

const = α0 + α1const−1 + α4const−4 + α5const−5 + (5)β0inct + β1inct−1 + β4inct−4 + β5inct−5 +γ0inft + γ1inft−1

Task: Estimate the above reduced model.

The next step of the sequential reducting is to attempt to move to the original DHSY function(omitting DOt variable), so to the form:.

∆4const = θ1∆4inct + θ2(const−4 − inct−4) + θ3∆∆4inct + θ4inft + θ5∆inft + εt (6)

It is difficult to immediately know the required restrictions. They will be better seen after sometransformations of equation (5). They become more evident if one adds and substracts const−4,β0inct−4 and γ1inft to and from the right hand side of the equation. After a little manipulationone can derive:

∆4const = α1const−1 + (α4 − 1)const−4 + α5const−5 + (7)β0∆4inct + (β0 + β4)inct−4 + β1inct−1 + β5inct−5 +(γ0 + γ1)inft − γ1∆inft + εt

If the following restrictions are imposed on model (7):

1. α1 = 0

2. α4 − 1 = −(β0 + β4)

3. α5 = 0

4. β1 = −β5

then model (7) becomes:

∆4const = β0∆4inct + (γ0 + γ1)inft + (α4 − 1)(const−4 − inct−4) (8)+β1∆4inct−1 − γ1∆inft + εt

Now when we add and substract β1∆4inct to and from right hand side of the equation (8) andrearrange the terms we get:

∆4const = (β0 + β1)∆4inct − β1(∆4inct − ∆4inct−1) (9)+(α4 − 1)(const−4 − inct−4) + (γ0 + γ1)inft

−γ1∆inft + εt

Finally it is easy to observe that equations (6) and (9) are equivalent, where:

• θ1 = β0 + β1

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• θ2 = α4 − 1

• θ3 = −β1

• θ4 = γ0 + γ1

• θ5 = −γ1

So by subsequent reductions of the most general model we obtained an original DHSY model.Lets get back to the estimation results of equation (5).Estimate the model:

lcons lcons(-1) lcons(-4) lcons(-5) linc linc(-1) linc(-4) linc(-5) infl infl(-1) c.

We can easily calculate long run consumption elasticities with respect to income and inflation.Use the command line:

scalar lrinc = (c(4) + c(5) + c(6) + c(7))/(1 − c(1) − c(2) − c(3)).

lrinc is equal 1.031464 (when You double click on its name the value is shown in the bottom leftcorner of the Eviews interface - on the gray bottom line) - what is the interpretation of that number?

Task: calculate and interpret long run elasticity of consumption with respect to inflation. Compareit with short run elasticity.

3 Granger causality testing

Variable xt causes (in Granger sense) variable yt if current values of yt can be better forecastedwith lagged values of xt than without them. Granger causality test is done by regressing yt onlagged values of yt and xt and jointly testing the significance of all lags of xt.

The null hypothesis is that all lags at xt are equal to 0 and therefore that xt does NOT Grangercause yt. Lets check if consumption is the cause of income and the other way arround.

Task: check causality between log consumption, log income and inflation

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The Granger Causality Test for might be very sensitive to the selected number of lags in theanalysis. One should be careful to choose the reasonable lag lengths. For example, when analyzingmonthly data, the reasonable lags can range from 1 to 12 or 24, etc.; for the quarterly data, thereasonable lag terms can range from 1 to 4, 8, 12, etc., and for the annual data, the reasonable lagterms should be less.

And here is the result.

As a result we reject the hypothesis that income does not Granger cause consumption (p −value < 0.000), but at 5% level we cannot reject the null hypothesis that consumption does notGranger cause income.

Task: test Granger causality between consumption and inflation.

4 Integration order/unit roots testing

Non-stationarity of time series is an important problem as it tremendously impacts the way inwhich data should be treated. Non-stationary data cannot be analyzed with traditional econometric

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techniques as in this case some basic assumptions are not met and correct reasoning on relationshipsbetween non-stationary time-series is impossible. That’s why it is so important to be able to teststationarity of the data. The easiest way is to draw a figure of the series and visually inspect it.Lets do it for consumption.

From the above figure it is obvious that consumption series is not stationary since it is increasedupward as time changes. However increasing pattern of the series is not always an indication ofnon-stationarity. The process might be stationary arround the trend line - it is called deterministicnon-stationarity. How can we transform the time series data from non-stationary to stationary?We need to work on differenciated series. But will taking first differences be enough to obtainstationarity (for economic time series most of the time taking first differences is enough)? Toanswer that question we need to run a formal test.

The most popular test of non-stationarity (unit root) is the (Augmented) Dickey-Fuller testwith null hypothesis of non-stationarity. In Eviews You can get results for three different variantsof the test: without constant and trend, with constant only and with constant and trend - themiddle variant is the most commonly used one.

Whenever conducting any statistical test remember that what one really needs to know is thenull hypothesis and the p-value (probability) of the test statistic. Tests are always done at someconfidence level α and whenever the p-value of the test statistic is lower than assumed confidencelevel - null hypothesis is rejected. When the opposite is true - p-value is higher than α the nullcannot be rejected (remember that one never accepts statistical hypothesis).

Unit root tests in Eviews can be run by choosing View/Unit Root Test...

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One needs to choose from wide range of tests, select if it will be conducted for variable inlevels, its first or second differences. Then one has to choose whether to include a constant and/ortrend and if the number of lags in test augmentation should be chosen automatically (based oninformation criteria) or by user.

And here are the results.

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As ADF test does not have strong power, one can verify the decision by using for exampleKPSS test. There is a difference in null hypothesis, which for KPSS tests assumes stationarity ofthe variable of interest.

The results differ from ADF test as KPSS does not provide a p-value, showing different criticalvalues instead. In this case we compare the test statistic value with the critical value on desiredsignificance level. If the test statistic is higher than the critical value, we reject the null hypothesisand when test statistic is lower than the critical value, we cannot reject the null hypothesis. Knowingthat we can observe that KPSS test results for consumption series are in line with the above ADFtest. We reject the null hypotesis, so consumption is not stationary.

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Next we will check if first differences of consumption are a stationary process. We can do thissimply by checking additional option in Unit Root test window.

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The ADF test result shows that first differences of consumption are still non-stationary, but thep-value is close to 5% level. To be 100% sure lets run KPSS test.

Task: Run KPSS test for first differences of consumption - interpret the results.

Lets plot these first differences. To do that we need to have differenciated values stored as anew variable. Therefore we generate new series of first differences by writing genr dcons =d(cons) in command window and then plot it.

Task: check the order of integration of income and inflation series for the whole sample. UseADF and KPSS test to confirm the conclusions.

5 Cointegration testing

Cointegration means the long run relationship between non-stationary time series. If there areseries (two or more) that are individually non-stationary (have to be integrated of the same order),but there exists their linear combination which is stationary - we call it cointegration.Visual inspection of the series expected to be cointegrated is always a good idea.

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We will test cointegration by two-step Engle-Granger procedure. Both variables have to beintegrated of the same order. So in the beginning we check the order of integration of consumptionand income (ADF test).

At 5% significance level we reject the hypothesis that consumption is not stationary(p−value =0.9927). After taking first differences consumption still seems to be non-stationary at 5% level(p − value = 0.0688). But as we checked it previously with KPSS test - finally we concludethat consumption is an I(1) process. We can also check that the conclusion of not rejecting thehypothesis in ADF test at 5% level comes from 7th lag in the ADF test, which significance wouldbe very difficult to interpret.Doing the same for income we get not rejection of non-stationarity for level of the variable (p −value = 0.9902), but after taking first differences we arrive at stationarity, so income is also an I(1)variable.Because both series are integrated of the same order, we can test cointegration between them, solets start Engle-Granger procedure.First we run a simple regression of one variable on the other saving the residuals (after estimatingequation select Proc/Make Residual Series...). The coefficients of the regression are parameters ofpossible cointegrating vector.

Second step of testing cointegration is checking the stationarity of saved residuals. If residualsare stationary, there is a cointegrating relationship between analysed variables. If residuals arenon-stationary - cointegration does not exist. Again ADF test is used here with null hypothesis ofnon-stationarity, so lack of cointegration.

The result of the test is that we cannot reject the hypothesis, that there is no cointegrationbetween consumption and income.

A note: the critical values of ADL test for cointegration (applied to residuals from regression)are different than those for an ADL test for data series. However often the latter are used as anapproximation of the first.

However if we conduct KPSS test its test statistic is equal to 0.065520 and is lower than anyof critical values provided, so we cannot reject the null hypothesis about stationarity of residuals.As KPSS test is more powerful finally we conclude that cointegration between consumption andincome exists.

Task: What is the cointegrating vector and how to interpret it?

Task: Test existence of cointegration between lcons and linc series.

6 Error Correction Model Estimation

After having found cointegration we can build an Error Correction Model including short term andlong term relationships. For a relationship between consumption and income the simplest ECMmodel on quarterly data would take a form:

∆4const = α0 + α1∆4inct + α2(const−4 − β1inct−4) + εt (10)

where α1 will be interpreted as short run reactions of consumption to changes in income andα2 will show the speed of return in direction of the the long run relationship in case of shocks.Therefore α2 is expected to be negative. Parameter β is a part of cointegrating vector and in fact

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instead of (const−4 − β1inct−4) we will use the residuals saved in the Engle-Granger cointegrationtesting procedure.

Lets estimate the model. First we generate new variables using command window:

genr d4cons=cons-cons(-4)genr d4inc=inc-inc(-4)

or

genr d4cons=d(cons,0,4)genr d4inc=d(inc,0,4)

And estimate the equation using resid01 (residuals from regression of consumption on income):

d4cons d4inc resid01(-4) c

The model seems to be incorrect as the coefficient at error correction term is positive ans statisticallysignificant (and one expects negative). However the value of Durbin Watson statistic is quite low(1.387427), suggesting positive first order autocorrelation of residuals. The easiest solution wouldbe to add autoregression to the model, so include among independent variables the lagged value(s)of consumption differential or lagged error terms (assuming autoregressive residual structure). Letsestimate below model:

∆4const = α0 + α1∆4inct + α2(const−4 − β1inct−4) + εt + α3εt−1 (11)

How to interpret the results? Is AR(1) term enough to get rid of autocorrelation in residuals? Doesthe error correction mechanism exist?

Task: estimate ECM model for log consumption and log income and interpret the results.

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Macroeconometrics - handout 2