taylor rule

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Data Analysis for ResearchM.Sc. Accounting and Finance, Banking and Finance,

Finance, Finance and Investment, Law and Finance

Davide Cafaro

Queen Mary University of London

05/03/2013

Davide Cafaro (Queen Mary University of London)

Data Analysis for Research 05/03/2013 1 / 22
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Estimating the simple Taylor rule

In this exercise, we want to test whether there exists a relationshipbetween the way in which monetary policy is detemined and stockmarket volatility.

We use the Taylor rule as the estimation framework and make use ofUK data ranging from 1990:1Q to 2010:3Q. The variables containedin the dataset are: RGDP, which is the UK real GDP, CPI, which is

the consumer price index, ftse, i.e. the stock market value andinterest is the interest rate, set by the Bank of England.

We start our analysis by estimating the following simple Taylor rule:

it = + (t t) + (yt y

t ) + t (1)

where it is the log of interest rate, t is the actual ination, t is thedesired level of ination, yt is the actual output and y

t is the

potential one.

Note that for convenience, we set t = 0. Therefore the interest rate

depends by the actual ination level and output gap only.Davide Cafaro (Queen Mary University of London)

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Estimating the simple Taylor rule

Note that the variables, which we need for our analysis have been

already calculated and included in the dataset.More specically, we generated it as the log of interest rate, t as theyear-on-year ination percentage rate and the output gap (yt yt ).In all cases we used the command Quck/Generate Series.

Remember that:

1 The year-on-year percentage change requires rst that you take the logCPI (pt) and, secondly, that you generate ination using the followingformula (to be written in EViews):

ination = 100 [pt p(t 4)]

2 In order to generate the ouput gap, we need to calculate the potentialoutput, yt . First, we take the log ofRGDP (we call this variable rt).Double click on this variable and when the spreadsheet opens, clickProc/Hodrick-Prescott. Generate the potential output (yt ) and,nally, the output gap as (yt y

t).

Davide Cafaro (Queen Mary University of London)

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Estimating the simple Taylor rule - Visual inspection

Our rst step consists of a graphical inspection of our data. Plotting

the variables of interest yields:

Ination and output gaps show a similar behavior. The interest rateremains stable over the period until the 2008, when we may assist to alarge decrease in it. It could be the consequence of the nancial crisis.

Davide Cafaro (Queen Mary University of London) Data Analysis for Research 05/03/2013 4 / 22
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Estimating the simple Taylor rule - OLS analysis

The further step consists of estimating equation (1) employing the

OLS estimator. Remember that we expect that both and bepositive. Results are reported in the following table:

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As expected, the coecients associated to ination and output are

positive and signicant. A 1% increase in ination leads to a 0.15%increase in the interest rate. Moreover a unitary increase in theoutput gap raises the interest rate by 0.17%.

Overall we may note that the R2 indicates that our model can explainabout the 24% of the variability of dependent variable. Moreover theF-statistic argues in favor of the correctness of our model, since itsassociated probability is smaller than 0.05. Finally, the DW statisticsis about 0.06. This indicates that our specication may suer from aproblem of serial correlation.

Taylor (1993) suggests that for conducive monetary policy should setboth and equal to 0.5. This implies that the output and theination gap own an equal weight in shaping monetary policy. A moresevere ination-targetting monetary policy is such that the weight forination is considerably larger compared to the one for output gap.



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To test the aforementioned restriction, we set a Wald test (from theouput window, click on View/Coecient diagnostics/Wald Test -Coecient restriction). The null is:

H0 : = 0.5 and = 0.5

HA : 6= 0.5 and 6= 0.5

The result of the Wald test is reported below:

We may note that the Wald test is extremely signcant. Therefore wecannot accept the null hypothesis stated above. This implies that

ination and output gap weight dierently in shaping monetary policy.Davide Cafaro (Queen Mary University of London) Data Analysis for Research 05/03/2013 7 / 22
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To have a visual impression regarding our model, we may considerhow it is performs in predicting the variability of the dependentvariable. To accomplish this objective, we plot the actual values alongwith the tted ones (from the ouput window, click on View/Actual,Fitted, Residuals/Actual, Fitted, Residuals Graph):

The immediate impression is that our model (green line)underestimate the actual data (red line) at the beginning of our series

while consistently overestimate it from the 2008 onwards.Davide Cafaro (Queen Mary University of London) Data Analysis for Research 05/03/2013 8 / 22
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According to the above graph, we may conclude that our model maybe misspecied and/or aected by a problem of omitted variables.

However, before addressing the above issues, we may want to checkthe stability of the relationship under investigation. More specically,the presence of some shocks may aect it.

For instance, the dotcom bubble at the beginning of 2001 may hadsome consequences on the way in which the monetary authorityshaped monetary policy. In order to disclose the presence of astructural break, we set up a Chow break test (from the window

output, select View/Stability Diagnostics/Chow Break test. Inthe window, which appears, write 2001 : 1, to underline that yoususpect that a break occurred at that point).



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The null hypothesis for this test is the following:

H0 : there is not a break

HA : thre is a break at 2001:Q1

The result of the test is the following:

We may note that all the reported statistics argue against theacceptance of the null hypothesis. Therefore, we should conclude thatthe dotcom bubble played a role in the way in which the monetarypolicy was set.



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Estimating the augmented Taylor rule - OLS analysis

According to the main literature, there exists the possibility that assetprice volatility aects the monetary policy by bringing moreinformation to the monetary authority.If we want to test this possibility, we need to slightly change theequation to be estimated as follows:

it = + (t

t

) + (yt y

t

) +n

k=1kstk+ t (2)

where stk is the year-to-year S&P500 index change, which we use asa proxy of the stock market volatility.The hurdle of the above estimation is to correctly set k, i.e. the

number of lagged values, which we should include in our model. Toaccomplish this task, we adopt a specic to general approach. Westart by including the rst lag. If it is signicant, we include thesecond lag. If both the rst and the second lags are signicant, weinclude the third lags and so on. We stop when some of the included

lags are not signicant.Davide Cafaro (Queen Mary University of London) Data Analysis for Research 05/03/2013 11 / 22
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The results from our rst estimations are the following:

The inclusion of the rst lag ofs improves the goodness of t of our

model. Instead of focusing on the R2, we look at the adjusted R2,since the latter is sensitive only to the inclusion of meaningfulregressors. We may note that it slightly increased from 0.21 to 0.22,this suggesting that our model can explain better the variability of thedependent variable.

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As far as the coecient associated to s(1) is concerned , we maynote that the associate t-statistic is equal to 1.84, with a p-value of

0.07. This means that we have only a mild evidence of signicance.Nonetheless, this is enough to claim that asset price volatility bringssome more information into the model. Note that consistently withthe theory, as the asset price volatility increases, the interest rate goes

up as well.Given this result, we re-estimate our model by including the secondlag of s. The results are the following:



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By including the second lag, we may note that s(1) loses itssignicance. Moreover, it displays a change in the sign. Moreover, thecoecient associated with s(2) is much more closer to the rejectionregion. Therefore, we may conclude that only the rst lag should be

included in our model.A second source of misspecication may come from the fact that ourmodel does not satisfy the man assumptions of the classicalregression model.

More specically, it may suer from heteroskedasticity and serialcorrelation of the residuals and at a small extent from normality ofthe residuals. Therefore, we carry out some tests to check our model.

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Estimating the augmented Taylor rule - Diagnostic tests

The rst test is the normality one (in the output window select

View/Residual Diagnostic/Histogram - Normality test).Remember that the Jarque Bera test has the null of normalitydistribution in the residuals. The result is reported below:

We may easily note that the Jarque Bera test strongly reject thehypothesis of normality in the distribution of the residuals. Actually,we could reach the same result by looking at the graph reported alongwith the statistic.



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The second test is the White test for heteroskedasticity (in the outputwindow select View/Residual Diagnostic/Histogram - Normalitytest). Remember that the test has the null of homoskedasticity in theresiduals. The result is the following:

The reported statistics indicate a clear rejection of the nullhypothesis. Therefore, our residuals are heteroskedastic.


E i i h d T l l Di i
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The third test, which we carry out, is the LM test for serial

correlation (View/Residual Diagnostic/Histogram - Serialcorrelation LM test. Remember to select 1 lag). The nullhypothesis for this test is that the residuals are not serially correlated.The results are the following:

Also in this case, we have enough evidence to reject the nullhypothesis. Actually, this result is consistent with the interpretationof the DW statistic reported in the estimation table. It is equal to0.11, this denoting the existence of positive serial correlation.


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Estimating the augmented Taylor rule - Robust S. E.

We may conclude by saying that our model suers from both serialcorrelation and heteroskedasticity.

We may overcome this problem, by re-estimate our model using theNewey-West option to obtain robust standard errors:

We may note that after re-estimating our model by using robuststandard errors, none of the variables enter the model signicantly,

although they show the expected sign.Davide Cafaro (Queen Mary University of London) Data Analysis for Research 05/03/2013 18 / 22

E i i h d T l l GMM
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Estimating the augmented Taylor rule - GMM

However, the previous results may be determined by the fact that theasset price volatility aects monetary policy indirectly.

Someone argued that the asset price volatility may aect both theoutput gap and the ination rate. Therefore, we may consider the

following auxiliary regressions:

t = t1 + (yt1 yt1) + st1 + t

(yt yt ) = t1 + (yt1 y

t1) +st1 + t

The results from the above regressions are reported in the followingslide.


E ti ti th t d T l l GMM
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E ti ti th t d T l l GMM
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According to the previous tables, it seems that stock marketvariability may aect both ination and output gap. Therefore, itmight be more appropriate to estimate our model using an IVprocedure, namely a GMM estimator:


Estimating the augmented Taylor rule GMM
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Right from the beginning, we may note that the probability associatedwith the J-statistic support the choice of our instruments.

We may note that the coecient associated with s(1) is positive,although not statistically signicant. This may be an evidence thatthe asset price volatility aects the interest rate only indirectly. Webased our conclusion on the results that we obtained in the previoustwo tables.

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