05 Diagnostic Test Of Clrm 2 3b445w

CLRM Assumptions & Diagnostics II Multicollinearity Model Specifictions Error

1

Topics    

Multicollinearity Under/ Overfitting a model Functional Form Errors on Measurement

07/04/15

Violation of the Assumptions of the CLRM 

Recall that we assumed of the CLRM disturbance : 1. E(ut) = 0 2. Var(ut) = 2 <  3. Cov (ui,uj) = 0 4. The X matrix is non-stochastic or fixed in repeated samples 5. ut  N(0,2)

MULTICOLLINEARITY

Nature of Multicollinearity

Multicollinearity 

Multicollinearity occurs when the explanatory variables are very highly correlated with each other.



Perfect multicollinearity  

Cannot estimate all the coefficients E.g. suppose x3 = 2x2

ˆ  ( X ' X ) 1 X ' y

and the model is yt = 1 + 2x2t + 3x3t + 4x4t + ut

Sources of Multicollinearity     

The data collection method employed Constraints on the model or in the population being sampled. Model specification Overestimated model Micronumerocity 

Essentially a sample (regression) phenomenon,

Estimation in the Presence of “High” Multicollinearity    

Although BLUE, the OLS estimator have large variance and covariances, making precise estimation difficult. R2 will be high but the individual coefficients will have high standard errors. The regression becomes very sensitive to small changes in the specification. Thus confidence intervals for the parameters will be very wide, and significance tests might therefore give inappropriate conclusions.

Impact of Multicollinearity to variance of OLS estimator 

Large variances and covariances of OLS estimators

Variance Inflating factor (VIF) Speed of increase of variance of OLS estimators 

VIF



Variances expressed with VIF

The Effect of Increasing r23 on

COV ( ˆ2 , ˆ2 ) and VAR ( ˆ2 )

Detection of Multicollinearity Scatterplot

Where C(dependent variable):comsumption expenditure, Yd: real disposable income, W; real wealth, I:real interest rate.

Detection of Multicollinearity (cont’)  



High R2 but few significant t ratios High pare-wise correlations among regressors (higher than 0.8). But it is not necessary that they be high to have collinearity in any special case.(e.g. x2t+x3t=x4t, three or more variables linear)

Eigenvalue and condition index. CI 



CI is between 10 and 30, there is moderate to strong multicollinearity CI exeeds 30 there is severe multicollinearity





Maximum eigenvalue Minimum eigenvalue



Tolerance and variance inflation factor 

If the VIF of a variable higher than 10, that variable said be highly collinear

Solutions to the Problem of Multicollinearity 

“Traditional” approaches, such as ridge regression or principal components. But these usually bring more problems than they solve.



Some econometricians argue that if the model is otherwise OK, just ignore it - Maybe Multicollinearity is not bad, if the objective is prediction Only.



The easiest ways to “cure” the problems are - drop one of the collinear variables - transform the highly correlated variables into a ratio - go out and collect more data e.g. - a longer run of data - switch to a higher frequency

Model specification error Under/ Overfitting a model Functional Form Errors on Measurement

“Economists are people searching in a dark room for a non-existent black cat; econometricians are regularly accused of finding one.” (Kennedy, Peter, A Guide to Econometrics, 1992, p.82)

Questions in Model Specification Error 1.

How does one go about finding the “correct” model?

2.

What types of model specification errors is one likely to encounter in practice?

3.

What are the consequences of specification error?

4.

How does on detect the specification errors?

5.

Having detected specification errors, what remedies can one adopt and with what benefits?

6.

How does one evaluate the performance of competing model/

Omission (Underfitting) of an Important Variable 





Suggested empirical model

yt  1   2 X 2t   3 X 3t  u1t

Used empirical model

yt   1   2 X 2 t  u 2 t

Error term from equ. (2)

u 2t   3 X 3t  u1t

(1) (2) (3)

Omission (Underfitting) of an Important Variable Consequences & Detection 

Consequences 

If the omitted variable is correlated with the included variable in regression (the correlation coefficient between the two variables is nonzero), then parameter (intercept and coefficient) are biased as well as inconsistent. 

  

2 var(ˆ 2 )   x22t

Detection 

where b32 

x x x

3t 3 t 2 2t

If the omitted variable is not correlated with the included variable in regression (the correlation coefficient between the two variables is zero), intercept still biased. Variance of the regression is incorrectly estimated. Variance of the intercept and other coefficients are biased. 



E (ˆ 2 )   2   3b32

Using t and F-test

2  var(ˆ2 )   x22t (1  r2,3 )

Inclusion (Overfitting) of an Irrelevant Variable 

Suggested empirical model



Used empirical model



Consequences

yt  1   2 X 2t  u1t

yt   1   2 X 2 t   3 X 3 t  u 2 t

  

(1)

(2)

Coefficient estimates will still be consistent and unbiased, Variance of the model is corrected estimated. The estimated variance will be inefficient (generally larger than those of the true model) 2  var(ˆ2 )   x22t

2 var(ˆ 2 )   x22t (1  r2,3 )

then,

var(ˆ 2 ) 1  var(ˆ2 ) 1  r23

ln (yt)   xt u 1 2 t

Adopting the Different Functional Form Log-linear, Linear-log, Double log 

Log-Linear: causes a 100xβ2% increase in yt.

a 1-unit increase in x 2t

y  (x2t)ut t 1 2ln

Adopting the Different Functional Form Log-linear, Linear-log, Double log (cont’) 

Linear-Log: a 0.01xβ2 increase in yt.

a 1% increase in x 2t causes

ln(y  (x2t)ut t) 1 2ln

Adopting the Different Functional Form Log-linear, Linear-log, Double log (cont’) 

Double Log: causes a β2 % increase in yt.

a 1% increase in x 2t

Adopting the Wrong Functional Form & Underfitting Detection  

Examination of Residuals. (e.g. total cost function) The Durbin-Watson d statistic.

Adopting the Wrong Functional Form & Underfitting Detection using Ramsey’s RESET test



Ramsey’s RESET test is a general test for mis-specification of functional form.



Essentially the method works by adding higher order of the fitted values 2 3 (e.g. yt , yt etc.) into an auxiliary regression:

Adopting the Wrong Functional Form & Underfitting Detection using Ramsey’s RESET test 

Process of Ramsey’s RESET test 1.

2.

2 Regress and obtain the estimated yt and yˆ t values

Obtain R2 from this regression. The test statistic is given by TR2 and is distributed as a . y   X  y ˆ 2   yˆ 3  u t



1

2

2t

3 t

3 t

t

2  So if the value of the test statistic is greater than a ( p  1) then reject the null

hypothesis that the functional form was correct.

yt  1   2 X 2t   3 yˆ t2   3 yˆ t3  ut

 2 ( p  1)

But what do we do if this is the case? 

The RESET test gives us no guide as to what a better specification might be.



One possible cause of rejection of the test is if the true model is yt  1   2 x2t   3 x22t   4 x23t  ut In this case the remedy is obvious.



Another possibility is to transform the data into logarithms. This will linearise many previously multiplicative models into additive ones:

yt  Axt e ut  ln yt     ln xt  ut

Parameter Stability Tests 

So far, we have estimated regressions such as yt = 1 + 2x2t + 3x3t + ut



We have implicitly assumed that the parameters ( 1, 2 and 3) are constant for the entire sample period.



We can test this implicit assumption using parameter stability tests. The idea is essentially to split the data into sub-periods and then to estimate up to three models, for each of the sub-parts and for all the data and then to “compare” the RSS of the models.



There are two types of test we can look at: - Chow test (analysis of variance test) - Predictive failure tests

The Predictive Failure Test 

 

Problem with the Chow test is that we need to have enough data to do the regression on both sub-samples, i.e. T1>>k, T2>>k. An alternative formulation is the predictive failure test. What we do with the predictive failure test is estimate the regression over a “long” sub-period (i.e. most of the data) and then we predict values for the other period and compare the two.

To calculate the test: - Run the regression for the whole period (the restricted regression) and obtain the RSS - Run the regression for the “large” sub-period and obtain the RSS (called RSS1). Note we call the number of observations T1 (even though it may come second). RSS  RSS1 T1  k Test Statistic   RSS1 T2 where T2 = number of observations we are attempting to “predict”. The test statistic will follow an F(T2, T1-k).

Predictive Failure Tests – An Example   

We have the following models estimated: For the CAPM  on BCA. 1980M1-1991M12 0.39 + 1.37RMt T = 144 RSS = 0.0434 1980M1-1989M12 0.32 + 1.31RMt T1 = 120 RSS1 = 0.0420 Can this regression adequately “forecast” the values for the last two years?

0.0434  0.0420 120  2 Test Statistic   = 0.164 0.0420 24 

Compare with F(24,118) = 1.66. So we do not reject the null hypothesis that the model can adequately predict the last few observations.

How do we decide the sub-parts to use? As a rule of thumb, we could use all or some of the following:

600 400 200

Sample Period

443

417

391

365

339

313

287

261

235

209

183

157

131

79

105

0 53

- Use all but the last few observations and do a predictive failure test on those.

800

27

- Split the data according to any known important historical events (e.g. stock market crash, new government elected)

1000

1

split the data accordingly to any obvious structural changes in the series.

1200 t)

- Plot the dependent variable over time and

1400

Value of Series (y



Testing the Normality Assumption 

Why did we need to assume normality for hypothesis testing?



The Bera Jarque normality test 

A normal distribution is not skewed and is defined to have a coefficient of kurtosis of 3.



The kurtosis of the normal distribution is 3 so its excess kurtosis (b2-3) is zero.



Skewness and kurtosis are the (standardised) third and fourth moments of a distribution.

Testing for Normality using Jarque Bera 

Bera and Jarque formalise this by testing the residuals for normality by testing whether the coefficient of skewness and the coefficient of excess kurtosis are tly zero.



It can be proved that the coefficients of skewness and kurtosis can be 3 expressed respectively as: b  E [u ] E[u4 ] 1 b2  2 3/ 2 2 and  2

 



 

The Bera Jarque test statistic is given by  b12  b2  3 2  2 W T   ~   2 24   6



We estimate b1 and b2 using the residuals from the OLS regression, u.

What do we do if we find evidence of Non-Normality? 

It is not obvious what we should do!



Using central limit theorem 

If Y1, ..., YN are independent and identically distributed random variables with mean μ and , then Y variance   Yi N σ2, and

Y μ ZN  σ N has a probability distribution that converges to the standard normal N(0, 1) as N → ∞ 

This theorem says that the sample average of N independent random variables from any probability distribution will have an approximate standard normal distribution after standardizing, if thea sample is sufficiently large.



Y ~μ, Nσ

2

N



Example of CLT 

Let the continuous random variable Y have a triangular distribution, with pdf:  2 y 0  y  1 f  y  

 0





otherwise

The expected value of Y is μ = E(Y) = 2/3 and its variance is σ2 = var(Y) = 1/18 Z N  Y  2 3 1 18 Then N

approaches the standard normal distribution as N → ∞

A Strategy for Building Econometric Models Our Objective: 

To build a statistically adequate empirical model which - satisfies the assumptions of the CLRM - is parsimonious - has the appropriate theoretical interpretation - has the right “shape” - i.e. - all signs on coefficients are “correct” - all sizes of coefficients are “correct” - is capable of explaining the results of all competing models

2 Approaches to Building Econometric Models     

There are 2 popular philosophies of building econometric models: the “specific-to-general” and “general-to-specific” approaches. “Specific-to-general” was used almost universally until the mid 1980’s, and involved starting with the simplest model and gradually adding to it. Little, if any, diagnostic testing was undertaken. But this meant that all inferences were potentially invalid. An alternative and more modern approach to model building is the “LSE” or Hendry “general-to-specific” methodology. The advantages of this approach are that it is statistically sensible and also the theory on which the models are based usually has nothing to say about the lag structure of a model.

The General-to-Specific Approach    

 

First step is to form a “large” model with lots of variables on the right hand side This is known as a GUM (generalised unrestricted model) At this stage, we want to make sure that the model satisfies all of the assumptions of the CLRM If the assumptions are violated, we need to take appropriate actions to remedy this, e.g. - taking logs - adding lags - dummy variables We need to do this before testing hypotheses Once we have a model which satisfies the assumptions, it could be very big with lots of lags & independent variables

The General-to-Specific Approach: Reparameterising the Model 

The next stage is to reparameterise the model by - knocking out very insignificant regressors - some coefficients may be insignificantly different from each other, so we can combine them.



At each stage, we need to check the assumptions are still OK.



Hopefully at this stage, we have a statistically adequate empirical model which we can use for - testing underlying financial theories - forecasting future values of the dependent variable - formulating policies, etc.

Questions?