Lecture 3 Dynamic Modelling

8/6/2019 Lecture 3 Dynamic Modelling

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Lecture 3

Stephen G Hall

Dynamic Modelling



The process of dynamic modelling has become such a central part of

Econometrics that it is worth treating it as a topic in its own right.

Dynamic modelling is a largely intuitive and simple process but it has become

surrounded by a specialised language, DGP, parsimonious encompassing,

conditioning, marginalising etc.

This lecture attempts to explain this jargon and why it is useful.



let xt be a vector of observations on all variables in period t,and let

Xt-1=(xt-1 ... x0), then the Joint probability of the sample xt, the DGP, may

be stated as,

)|( X x D 1-t t t

t

=1t

54 :

Where is a vector of unknown parameters.

The Philosophy underlying this approach is that all models are

misspecified. The issue is to understand the misspecification and to build

useful and adequate models.

5



The process of model reduction consists principally of the following four

steps.

1. Marginalise the DGP.We select a set of 'variables of interest' and

relegate all the rest of the variables to the set which are of no interest.

2. Conditioning assumptions. Given the choice of variables of interest we

must now select a subset of these variables to be treated as

endogenous

3. Selection of functional form. The DGP is a completely general

functional specification and before any estimation can be undertaken a

specific functional form must be assumed.

4. Estimation. The final stage involves assigning values to the unknown

parameters of the system, this is the process of econometric

estimation.



given the general DGP it is possible to represent the first two stages in

the model reduction process by the following factorisation.

); Z ,Y | Z ( C ); Z ,Y |Y ( B ); X |W ( A

= ); X | x( D

1-t 1-t t t t 1-t t t t t t

1-t t t

K FE

5

These steps are all crucial in the formulation of an adequate model.

If the marginalisation is incorrect then this implies that some important

variable has been relegated to the set of variables of no interest.

If the conditioning assumptions are incorrect then we have falselyassumed that an endogenous variable is exogenous.

If the functional form or estimation is invalid then obvious bias results



Exogeneity

Conditioning is basically about getting the determination of exogeneity

right, there are three main concepts of exogeneity

Weak exogeneity

Z is weakly exogenous if it is a function of only lagged Ys and the

parameters which determine Y are independent of those determining Z.

Strong exogeneity

here in addition we assume that Z is not a function of lagged Y. this is

weak exogeneity plus non granger causality

Super exogeneity

here in addition we assume that the parameters which determine Z are

independent of the parameters which determine Y.

Weak exogeneity is needed for estimation. Strong exogeneity is needed

for forecasting. Super exogeneity is needed for simulation and policy

analysis.



Before the development of cointegration the dynamic modelling approach

in practise began from a general statement of the DGP suitably

marginalised and conditioned,

u+ X B+Y +=Y t i-kt ki

n

0=i

m

1=k

i-t i

n

1=i

0t §§§EE

This general form (the ADL) may be reparameterised into many differentrepresentations which are all either equivalent or are nested within it as

restrictions. eg the Bewley transformation, the common factor restriction.

A particularly useful for is the Error Correction Mechanism (ECM)

u )+Y +( -

Y Y

t 1-kt k

1k

1-t *0

i-kt *

ki

1-n

0i1k

i-t *i

1-n

1i

t +

FEK

FE (((



The basic idea in dynamic modelling is that the General model should be

set up in such a way that it passes a broad range of tests, in particular

that it should have constant parameters and a `well' behaved error

process.

The model is then reduced or simplified applying a broad range of tests at

each stage to try and find an acceptable parsimonious (minimum number of parameters) representation.

This is the process of model reduction.

In practise the real issue is to understand the tests used.



The general F test.

The general test used for testing a group of restrictions is the F-test, this

tests any restricted model against a less restricted model.

¹ º ¸©ª̈¼½

»¬«

mk -T

RSS RSS - RSS =k)-T F(m,1

12

T- sample sizek- number of parameters in unrestricted model

m- number of restrictions.



The lagrange multiplier test for serial correlation

if u is the residual from an OLS regression then perform

X +

u=u i-t i

m

1=it &§K

Then an LM test of the assumption that there is no serial correlation up to

order m is given by LM(m)=TR2



Instrument validity test

when estimating an IV equation we should test that the instruments are

weakly exogenous, this may be done by performing the following

auxiliary regression

EW ut !

whereW is a set of variables which includes both the independent

variables in the equation and the full set of instruments. The test is (T-k)R2

which is , where r is the number of instruments minus the number of endogenous variables in the equation,(r)

2

G



The Box-Pierce and Ljung-Box test

This is based on the correlogram and is a general test of mth order serial

correlation

r )i-( Q

r Q

2ii-

1i

*

2i

1i

+2)( §

§

This is again a chi sq test with m degrees of freedom



ARCH test

u+u2i-t i

1i

02t EE §

TR2 from this regression is a test of an autoregressive variance process of order

m. It again is a chi sq test with m degrees of freedom



Parameter Stability

The Chow test is a test of parameter constancy which is a special form of Ftest

¹ º

¸©ª

¨

¼½

»

¬

«

k

2k -

)SS +SS (

)SS +SS ( -SS

HOW 21

21

which is distributed as F(k,T-2k).



In practise we tend to give greater weight to recursive estimation. This is

a series of estimates where the sample size is increased by one period in

each estimation. If we define as the estimate of the vector of parameters based on the period 1 to t. then we can define the recursive

residuals as

F t

X -Y =v t 1-t t t Fwe can then standardise these for the degrees of freedom so that

) N(0,d / vw2

t t t W ~

now they have the same properties as the OLS residuals except that they

are not forced to sum to zero and they are much more sensitive to model

misspecification.



Formal tests based on the recursive residuals are;

The CUSUM test,

w(1/s)USUM i

t

1+k i

t §

where s is the full sample estimate of the standard error.

¹ º

¸©ª

¨

¹ º

¸

©ª

¨

§§ wwUSUMSQ

2

i1+k i

2

i

t

1+k it /

But in practise plots of the recursive residuals and parameters are often

much more informative.

THE IMPORTANCE OF GRAPHS IS CRUCIAL

The CUSUMSQ test



Testing Functional Form

The Ramsey RESET test checks the possibility of higher polynomial

terms

Y + X =u1+i

t i

m

1=i

t t ÖEK §d

again this is an LM test based on TR2



Testing for Normality

Normality of the residuals is an important property in terms of justifying

the whole inference procedures, typical test is theBera-Jarque test

(2) )3-(EK

24+

SK 6 J

222

G ~

¼½

»

¬

«

where SK is the measure of skewness and EK is the measure of excess

kurtosis



Encompassing

This is a general principal of testing which allows us to interpret and

construct tests of one model against another

A model (M1) encompasses another model (M2) if it can explain the

results of that model.

Many standard tests (eg F or LM) can be interpreted as encompassing

tests.

Parsimonious encompassing

a large model will allays encompass a smaller nested model, this is not

interesting. If a smaller model contains all the important information of a

larger model this is important and we then say that it parsimoniouslyencompasses the larger model

Variance encompassing

asymptotically a true model will always have a lower variance than a false

model, so the finding of a smaller standard error is evidence of variance

encompassing.



Why Variance encompassing is better than using the R2

The R2

statistic is not invariant to the way we write an equation

t t t t uY Y !1E F

If Y is trended it will generally have very high R2

t t t t uY X Y !( 1)1(E F

Exactly the same equation, just a reparameterisation, exactly the same errors

BUT a completely different R

2

as we have changed the dependent variable. Theerrors and the error variance are unchanged.



Example

Davidson, Hendry, Srba and Yeo

(DAISY)



Note:strong seasonality, upward near

proportional trend



Note: Annual changes: consumption much

smoother than income



NOTE: The APC is not constant but changes

systematically



Note: The seasons are

different, so seasonality is

important



Note: notice the scale, changes in

income much bigger than consumption



Note: the seasonal pattern is changing

through time.



Start by considering the best existing models and what is

wrong with them

Older Hendry model, very

low long run MPC

LBS Ball model, Low

long run MPC no

seasonality

Wall model; no long run at all



Difference model or ECM, the first difference is only valid

under testable restrictions



General to Specific: tests on an invalid model are themselves

invalid

Insignificantsignificant



So start from a general model and nest down to a specific one

But final model

has no long run

and fails to

forecast



So set up an ECM to impose the long run proportionality

Seasonally adjusted data

BUT: both fail to forecast

so back to the beginning



A possible missing variable; inflation may explain the

movement in the APC



So start again, and eventually 2 models one without a long

run one in ECM form

ECM passes

the forecast

test



Final validation, out of sample forecasting performance

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Lecture 3 Dynamic Modelling