ARMA Estimation

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OUTLINE INTRODUCTION ESTIMATION ASYMPTOTIC PROPERTIES IDENTIFICATION DIAGNOSTIC CHECKING FORECASTING MODEL ID

MODEL IDENTIFICATION, ESTIMATION AND

DIAGNOSTIC CHECKING

Vlayoudom Marimoutou

AMSE Master 2

Autumn 2013

December 11, 2013

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1 INTRODUCTION2 ESTIMATION OFARMA(p, q) PROCESSES

Estimation ofAR(p)processes

estimation of anAR(p)for a fixed (and known)pEstimation ofAR()processes

Estimation ofARMA(p, q)process: Maximum likelihood estimationLikelihood function forAR processes

Exact versus conditional MLE

likelihood function forMA process

Numerical optimization of the log likelihood3 ASYMPTOTIC PROPERTIES OF ML ESTIMATORS AND

INFERENCE4 IDENTIFICATION OF AN ARMA PROCESS: SELECTINGp

ANDq

Information criteriaOther model selection approaches5 DIAGNOSTIC CHECKING6 FORECASTING

The MSE optimal forecast is the conditional mean

Forecast evaluation

Relative evaluation:Diebold-Mariano

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1 INTRODUCTION

2 ESTIMATION OF

ARMA(p, q) PROCESSES

3 ASYMPTOTICPROPERTIES OF ML

ESTIMATORS AND

INFERENCE

4 IDENTIFICATION OF ANARMA PROCESS:

SELECTINGp ANDq

5 DIAGNOSTIC CHECKING

6 FORECASTING

7 MODEL IDENTIFICATION

IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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Suppose you want to fit anARMA(p, q)model to aunivariatestationaryprocess {Xt}.The first thing will be to set values for p and q. We will call this step

identification.

Next, we would like toestimatethe correspondingARMA(p, q). Themost popular method for estimatingARMAprocesses is Maximum

Likelihood (ML).

ARprocesses can be also estimated by OLS. It can be shown that this

approach is asymptotically equivalent to ML. This is very convenient

because OLS estimator have closed analytical expressions and are

straight forward to compute.

The final step will be to check whether the residuals of th estimated

model verify the assumptions that have been imposed on the

innovations.

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1 INTRODUCTION

2 ESTIMATION OF


3 ASYMPTOTICPROPERTIES OF ML

ESTIMATORS AND

INFERENCE


SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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ESTIMATION OFAR(p) PROCESSES

.

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O I O C O ES A O AS O C O S I CA O D AG OS C C C G FO CAS G MO


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Assume for the time being that(p, q)are known. We will consider firstestimation ofAR processes (its simpler) and then we will move to

estimation of generalARMA(p, q)processes.

Here we distinguish two cases:

Xtfollows anAR(p)for some finitep

Xtfollows anAR()process.

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Let {Xt}be a stationaryAR(p)process withp < , i.e.

Xt=0+1Xt1+...+pXtp+t

wheretis a zero mean innovation (either w.n, m.d.s. or i.i.d.) with variance2.

It is possible to estimate=

01. . . p)

by applying the standard OLS

formula and the resulting estimator is

T- consistent and asymptotically

normal

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We now provide a sketch of the proof for the simplest case. Let {Xt} be anAR(1)process given by

Xt=Xt1+t

wheretis am.d.sand

||


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Consistency

olsp

Xt1tX2t1

p 0

Under the previous assumptions,X2t is also stationary and ergodic, thus by

LLN,

X2t1T

p var(Xt1) = 2

(1 2)As for the numerator, notice that iftis am.d.s, so is

Xt1t. Therefore, a

LLN also applies to the numerator and Xt1tT

p

E(X

t1

t) =0. Then

ols=+ op(T)

Op(T)=+

op(1)

Op(1)=+op(1)

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O


Asymptotic normality

T12

ols

=T1/2Xt1t

T1

X2t1

=

T12Xt1tvar(Xt)

+op(1)

By the CLT for m.d.s

T12 Xt1t d N0, var(Xt)2

Thus

T12 ols

d

N0,

2

var(X

t)= N0, (1

2)Remark: What happens with the distribution ofolsif 1?

A similar proof can be established for a general stationaryAR(p)process.

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AR(p)(for finitep) are easier to estimate thanMA processes since OLScan be applied.

We know that invertibleARMAprocess can be written as anAR()process.

Berk showed that if anAR(k)process is fitted toXt, wherektends to infinitywith the sample size, it is possible to obtainconsistent and asymptotically

normally distributedestimators of the relevant coefficients.

More explicitly,khas to verify two conditions:

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an upper bound conditionsk3/T

0 (that says thatkshould not

increase too quickly),

and a lower bound one,T1/2

j=k+1i c =0 (that says thatkmustnot increase too slowly).

How to choosekin practice?

The above mentioned conditions do not help much in choosing the

appropriatekin applications. Ng an Perron (2005) have shown that the

value ofkchosen by the AIC and the BIC does not verify the lower

bound condition. This implies that the estimates are consistent but not

asymptotically normal

Kuersteiner (2005) has proved that the general to specific model

selection verifies both the upper and the lower bound conditions. Then,

the resulting estimates are consistent and asymptotically normal.

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ESTIMATION OFARMA(p, q) PROCESS: MAXIMUM LIKELIHOOD ESTIMATION

Let {Xt} be anARMA(p, q)process

p(L)Xt=+q(L)t

and let=

1,...,p, 1,...,q, , 2

be the vector containing all the

unknown parameters. Suppose that we have observed a sample of size

T, (x1,x2,...).

The ML approach will amount to calculate the joint probability density

of(XT,...,X1)

fXT,XT1,...,X1 (xT,xT1,...,x1; ) (1)

which might be loosely interpreted as the probability of having

observed this particular sample.

The Maximum likelihood estimator (MLE) ofis the value thatmaximizes(1), i.e, the probability of having observed this sample.

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ESTIMATION OFARMA(p, q) PROCESS: MAXIMUM LIKELIHOOD ESTIMATION

The first step is to specify a particular distribution for the white noise

innovations of the process,t. Typically, it will be assumed thattis a

Gaussian white noise,t i.i.dN(0, 2).This assumption is strong but when Gaussianity does not hold , the

estimator computed by maximizing the Gaussian likelihood has, under

certain conditions, the same asymptotic properties as if the process

were indeed normal. In this cases, the estimator of the parameters of a

non Gaussian process computed on a Gaussian likelihood is calledQuasi or Pseudo MLE

The second step would be to calculate the likelihood function(1). Theexact closed form of the likelihood function of a generalARMA

processes is complicated. In the following we will illustrate how the

likelihood is computed for simpleAR andMA processes.The final step is to obtain the values ofthat maximize the loglikelihood. Unless the process is a pureAR process (in which cas OLS

can be applied), numerical optimization procedures should be applied

in order to obtain the estimates.

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LIKELIHOOD FUNCTION FORAR PROCESSES

Let {Xt} be anAR(1)process

Xt=c+Xt1+t

where ||


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As for theX3, the distribution ofX3conditional onX2 =x2andX1 =x1 is

fX3|X2,X1 (x3|x2,x1; ) = 1

22e

(x3cx2)2

22

The joint distribution of theX1,X2 andX3 can be written as

fX3,X2,X1 (x3x2,x1; )

= fX3|X2,X1 (x3|x2,x1; )fX2|X1 (x2|x1; )= fX3|X2,X1 (x3|x2,x1; )fX2|X1 (x2|x1; )fX1 (x1; )

Furthermore, notice that the values ofX1,X2,...,XT1 matter forXtonlythrough the value ofXt1, and then

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Taking logs

L() =log(fX1 (x1; )) +T

t=2logfXt|Xt1 (xt|xt1; ) (7)

and L is called the log-likelihood function.

Clearly, the value ofthat maximizes(6)and(7)is the same but themaximization problem is simpler in the latter case, and the the log likelihood

function is always preferred.

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The next step would be to compute the value offor which the exact loglikelihood in(2)is maximized. This amounts to deriving the log likelihood

and equating the first derivatives to zero. The result is a system of non linearequations onand the sample for which there is no close solution in termsof(x1,...,xT. Then, iterative numerical procedures are needed to obtain.

And alternative procedure is to regard the value ofy1 as deterministic and

then

logfXT,XT1,...,X1 (xT,xT1,...,x1; )

=logfXT|XT1 (xT|xT1; )fXT1|XT2 (xt1|xt2; )...fX1 (x1; )

=

T

t=2

log fXT|XT1 (xT|xT1; )=

T 1

2

log

22

Tt=2

(xt c xt1)2

22

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Notice that the conditional MLE ofc and are obtained by minimizing

Tt=2

(xt c xt1)2

It follows that maximization of the log likelihood is equivalent to theminimization of the sum of squared residuals. More generally, the

conditional MLE for anAR(p)process can be obtained from an OLSregression ofyton a constant andp of its own lagged values. In contrast to

exact MLE, the conditional maximum likelihood are trivial to compute.

Moreover, if the sample sizeTis sufficiently large, the first observationmakes a negligible contribution to the total likelihood provided ||


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The conditional MLE of the innovation is found by differentiating (9) with

respect to2 and setting the result equal to zero. It can be checked that

2 =

Tt=2

(xt c xt1)2T 1 ,and in general, the MLE estimator of2 of anAR(p)process is given by thesum of squared residuals over(Tp)

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LIKELIHOOD FUNCTION FORMA PROCESS

Let {Xt} be a gaussianMA(1)processXt=+t+t1

withtisiid N(0, 2). Let=

,,2

be the population parameters to be

estimated. Then,Xt|t

N+t1, 2andfXt|t1 (xt|t1) =

122

e

(xtt1)

2

22

It is not possible to compute this density sincet1is not directly observed

from the data. Suppose that it is known that0 =0. Then, all the sequenceof innovations could be computed recursively as

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1=x

1

2 =x2 x1

...

t=xt xt1Then the conditional density is given by

fXt|Xt1,...,X1 (xt|xt1,...,x1, 0 =0; ) =fXt|t1 (xt|t1)

= 1

22e

2t22

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while the conditional log likelihood is

L() = T2

log

22 2t

22

The conditional log likelihood function is a complicated non linear function

and, in contrats to theAR case, they should be found by numerical

optimization.

If is far from 1 in absolute value, then the effect of imposing0 =0 willquickly die out. However, if is over 1 in absolute value, the error ofimposing this restriction accumulates over time. Then, if the output of

estimating is greater than1 should discard the results. The numericaloptimization should be attempted again with the reciprocal of used asstarting value.

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NUMERICAL OPTIMIZATION OF THE LOG LIKELIHOOD

Once the log likelihood has been computed, the next step is to find the valuethat maximizes L(). With the exception of pureAR processes, for whichclosed analytical expressions of the estimators are available, numerical

optimization procedures should be employed to obtain the estimates. These

procedures make different guesses for, evaluate the likelihood at thesevalues and try to infer from these there the valuefor whichis largest. SeeHamilton, section 5.7 for a description of these methods.

The search procedure may be greatly accelerated if the optimization

algorithm begins with parameter values which are close to the optimum

values. For this reason, simple preliminary estimates ofare often

employed to begin the search. See Brockwell and Davis, Section 8.2-8.4 fora description of these preliminary estimators.

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1 INTRODUCTION

2 ESTIMATION OF


3

ASYMPTOTIC

PROPERTIES OF ML

ESTIMATORS AND

INFERENCE


SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8

R CODE

9 REFERENCES

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ML estimators have very good asymptotic properties.

Under certain conditions it can be shown that they are consistent,asymptotically normal and efficient, since the variance covariance

matrix equals the inverse of Fischer information matrix.

Let {Xt} be anARMA(p, q)process,0be the vector containing the trueparameter values andis the MLE of. Assuming that neither0 norfallson the boundary of the parameter space then

T

0

d N0,I1where

Iis the Fischer Information matrix

I= E

2L()

|=0

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An estimator of the variance covariance matrix is

I= T1

2L()

|=

which is often calculated numerically. Another popular estimator of the

Fischer information matrix is the so-called outer product estimator

I= T1T

t=1

ht()

ht()

whereht() =logf(y

t|y

t1,y

t2,...; )/

|=denotes the vector of

derivatives of the log of the conditional density of thet thobservation withrespect to the elements of the parameter vector, evaluated at.

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1 INTRODUCTION

2 ESTIMATION OF


3 ASYMPTOTIC

PROPERTIES OF ML

ESTIMATORS AND

INFERENCE


SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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How to selectp andq?

ACFandPACFare of help but their usefulness is limited in the general

ARMA(p, q)case.In this section we will see other ways of selecting these values: model

selection criteria

If you assume that your model contains an infinite number of

parameters or that the type of models you are considering for

estimation do not include the true model, thegoalwill be to select onemodel that best approximates the true model from a set of finite

dimensional candidate models. In large samples, a model selection

criteria that chooses the model with minimum mean squared error

distribution is said to be aasymptotically efficient

Many researchers assume that the true model is of finite dimension andis included in the set of candidate models. The goal in this case is to

correctly choose the true model from the list of candidates. A model

selection criteria that identifies the correct model asymptotically with

probability 1 is said to be consistent

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INFORMATION CRITERIA


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Information criteria are mechanisms designed for choosing an

appropriate number of parameters to be included in the model. Byminimizing some particular distances between the true and the candidates

models (the Kullback-Leidler discrepancy, see Gourieroux-Montfort), it is

possible to arrive to expressions that usually have the general form:

IT(k) =log(2k) + k

C(T)

T penalty term

(8)

Some popular IC are:

1 Akaike:C(T) =2

2 Schwartz or Bayesian (BIC):C(T) =log T

3 Hannan and Quinn (HQ):C(T) =2 log(log T)

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The numbers of parameterskto be included in the model is chosen as

k= arg minkm

IT(k)

wherem is some pre-specified maximum numbers of parameters

Remark 1The AIC is not consistent, in the sense that it might not choose

asymptotically the correct model with probability one. It tends to

overestimate the number of parameters to be included in the model.

However, this does not mean that this method is useless. It minimizes the

MSE for one-step ahead forcasting

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Remark 2The BIC and the HQ criteria are consistent that is

limT

P(k=k0) =1 (9)

In fact, the consistency result (9) holds for any criterion of the type (8)

withlimT C(T)/T=0 and limT C(T) =

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OTHER MODEL SELECTION APPROACHES


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General to specific criterion (Ng and Perron, 1995). This methodamounts to

1 Set the maximum numbers of parameters to be estimated. For instance, an

AR(k)model withk=7.2 Estimate the "general" model, (anAR(7)in this case).

3 Test for the statistical significance of the coefficients associated with thehigher order lags - with either a tor anFtest (yt7 in this case).

4 Exclude the nonsignificant parameters, reestimate the smaller model and

test again for significance. Repeat until all the remaining estimates are

significantly different from zero.

Bootstrap model selecting techniques.

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1 INTRODUCTION

2 ESTIMATION OF


3 ASYMPTOTIC

PROPERTIES OF ML

ESTIMATORS AND

INFERENCE

4

IDENTIFICATION OF ANARMA PROCESS:

SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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Typically, the goodness of fit of a statistical model is judged by

comparing the observed data with the values predicted by the estimated

model.

If the fitted model is appropriate, the residuals should behave in a

similar way as the true innovations of the process.

Thus, we should plot the residuals of the estimated model to check whether

they verify the main assumptions: they are zero mean, homoskedastic (with

constant variance) and with no significant correlations. If the graph shows a

cyclic or trended component or a non- constant variance, it can be

interpreted as a sign of misspecification.

The autocorrelation function of these residuals should not display any

significant correlations. Let {et} be the sequence of residuals, given byet=Xt Xt

where Xtare the fitted values.

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Then, the autocorrelation function is given by

e(h) =

Tht=1(et e)(et+h e)T

t=1(et e)2, h= 1, 2,...

Assume first that the true parameter values were known. Then,et=t. In

this case, we know (see Chapter 1) that Ted

N(0,IH), where = ((1),...,(H))and IHis theHHidentity matrix, then the nullhypothesis that the firstHautocorrelations are non significant can be tested

using the Box-PierceQ statistic:

T

Hi=1

2(i) d

2H (10)

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However, in practice the values of the parameters are unknown and they

should be estimated. The fact that the parameters employed to computeetare not the true ones but are estimates has an impact on the asymptotic

distribution.

More specifically, the asymptotic variance of the sample autocorrelations isnot the identity matrix anymore. Hence, in this case (10) no longer holds.

However, under certain assumptions it is still possible to use the sample

autocorrelations for diagnosis of the model. The following proposition

presents the distribution of the sample autocorrelations of the sample

residuals, when the parameters of the model are estimated.

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PROPOSITION

Suppose that Yt=Xt+t, t= 1,..., T, where Ytandtare scalar and Xt

andare k 1vectors. If{Yt,Xt} are jointly stationary and ergodic,E(XtXt)is a full rank matrix,

E(t|t1, t2, ...,Xt,Xt1,...) =0

and

E

2t|t1, t2,...,Xt,Xt1,...

= 2 >0

then

Te

d

N(0,IH

Q)

where qjk(the jk element of the HH matrix Q) is given by

E(Xttj)E(XtX

t)1

E(Xttk) /2

Proof. see Hayashi, p165

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Further, ifXtis anARMA(p, q)process and the model is correctly specified,it can be shown that the Box Pierce Q -statistic for diagnosting checking

when the parameters are estimated is approximately distributed as

Qe=T

Hi=1

2e (i) d 2(Hpq)

The adequacy of the model is rejected at level if

Qe=Ti=1

H2e (i)> 21(Hpq)

Finally, Ljung and Box (1978) suggestreplacing Qeby

Qe= n(n+2)Hi=12e (i)

n isince they claim that the statistics offers a better approximation to the2

distribution.

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1 INTRODUCTION

2 ESTIMATION OF


3 ASYMPTOTIC

PROPERTIES OF ML

ESTIMATORS AND

INFERENCE

4

IDENTIFICATION OF AN

ARMA PROCESS:

SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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Ah-step ahead forecast,xt+h|t, is designed to minimize expected loss,conditional on timetinformation. Examples of widely-used lossfunctions:

MSE:xt+h xt+h|t

2

MAD:xt+h xt+h|t

Quad-Quad: 1xt+h xt+h|t

2+2I[xt+hxt+h


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Letxt+h =Et[xt+h]

Letxt+h be any other value

Et

(xt+h xt+h)2

= Et

xt+h xt+h

+xt+h xt+h

2=Etxt+h x

t+h

2+2 xt+h x

t+h xt+h xt+h+ x

t+h

xt+h

2

=Vt[xt+h] +2Et

xt+h xt+h

xt+h xt+h

+Et

xt+h xt+h

2=Vt[xt+h] +2

xt+h xt+h

Et

xt+h xt+h

+Et

xt+h xt+h

2=Vt[xt+h] +2 xt+h xt+h .0+Etx

t+h

xt+h2

=Vt[xt+h] + (xt+h xt+h)2

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OUTLINE

INTRODUCTION

ESTIMATION

ASYMPTOTIC PROPERTIES

IDENTIFICATION

DIAGNOSTIC CHECKING

FORECASTING

MODEL ID

THE M SE OPTIMAL FORECAST IS THE CONDITIONAL MEAN


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MSE optimal forecast for anAR(1):

xt=1xt1+t

Et[xt+1] =Et[1xt+t+1] =1Et[xt] +Et[t+1] =1xt+ 0

Et[xt+2] =Et[1xt+1t+2] =1Et[xt+1] +Et[t+2] =1(1xt) +0= 21xt+ 0

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OUTLINE

INTRODUCTION

ESTIMATION

ASYMPTOTIC PROPERTIES

IDENTIFICATION

DIAGNOSTIC CHECKING

FORECASTING

MODEL ID

FORECAST EVALUATION


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Two primary criteria to evaluate forecasts:

Objective

Relative

Objective: Mincer Zarnowitz regressions

xt+h =+xt+h|t+t

H0 : = 0, =1;H1 : =0 =1Use any test: Wald, LR, LM

Can be generalized to include any variable available when the forecast

was produced

xt+h =+xt+h|t+zt+t

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FORECAST EVALUATION


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H0 : = 0, =1, =0;H1 : =0 =1 j=0

ztmust be in the timetinformation set

Important when working with macro data

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RELATIVE EVALUATION:D IEBOLD-M ARIANO


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Two forecasts,xAt+h|tandxBt+h|t

Two losses

lAt = (yt+h yAt+h|t)2 andlBt = (yt+h yBt+h|t)2

Losses do not need to be MSE

if equally good or bad,E[lAt] =E[lBt] or E[l

AtElBt] =0

Define=lA

t lB

t

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RELATIVE EVALUATION:D IEBOLD-M ARIANO


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Implemented as at-test thatE[t] =0

H0 :

E[t] =

0;HA

1 :E

[t]0

Composite alternative

Sign indicates which model is favored

DM=

V[

]

One complication: {t} cannot be assumed to be uncorrelated, so amore complicated variance estimator is required

Newey-West covariance estimator:

2 = 0+2

Ll=1

1 l

L+1

l

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1 INTRODUCTION

2 ESTIMATION OF


3 ASYMPTOTIC

PROPERTIES OF ML

ESTIMATORS AND

INFERENCE

4 IDENTIFICATION OF AN

ARMA PROCESS:

SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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Identifying a model forytrefers to the methodology for selecting

The appropriate transformations for obtaining

stationarity (such as variance stabilization

transformations and differencing)

values forp, qandd(the integration order)

deciding whether a deterministic component shouldor should not be included in the model.

The followings steps should be followed to identify the model:

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Step 1. Plot the data and choose the proper transformations.

The plot usually shows whether the data contains a trend, a seasonal

component, outliers, non constant variances, etc. then, it might suggest

some proper transformations.

The most commonly used transformations are variance stabilizing

transformations, typically, a logarithmic transformations or, more

generally, a Box Cox transformation, and/or differencing.

Always apply the variance stabilizing transformations before taking any

number differences.

Step 2. Compute and examine the sample ACF and sample PACF of the

(variance-stabilized) process.

If these functions decay very slowly, this would be a sign of non stationarity.

Unit root tests should also be used at this stage. Typically, the number of

differences would be 0, 1 or 2.

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Step 3. Compute the ACF and PACF of the transformed variable to identify

pandq.

Identifying the order of sampleAR or MA polynomials is, in theory, easy

with the table above. However, it is more difficult to identify the orders of an

ARMAprocess. In these cases, other model selection mechanisms, such as

information criteria (see below) can be implemented.

Step 4. Test the deterministic trend term whend>0.

Ifd>0 and a trend in the data not suspected should not be included.However, if there is a reason to believe that a trend should be include, one

can include this term in the model and the, discard it if the coefficient is not

significant.

For some interesting exemples, see Wei, Chapter 6 and Brockwell and

Davies, Chapter 9.

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ESTIMATION


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Once the appropriate transformations have been performed, the resulting

process is stationary (basically, variance stabilizing transformations and/or

differencing)

At this stage, the estimation techniques developed for the stationaryARMA

case are applicable to the resulting stationary process.

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1 INTRODUCTION

2 ESTIMATION OF


3 ASYMPTOTIC

PROPERTIES OF MLESTIMATORS AND

INFERENCE


ARMA PROCESS:

SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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Download theR Library "Time Series"


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A first step in analysing time series is to examine the autocorrelations (ACF)

and partial autocorrelations (PACF).R provides rhe functionsacf()and

pacf()for computing and plotting ofACFandPACF

sim.ar


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Diagnostic Checking

A first step in diagnostic checking of fitted models is to analyze the residuals

from the fit for any signs of randomness. Rhas the functiontsdiag(), whichproduces a diagnostic plot of fitted time series model

tsdiag(fit)

it produces output containing a plot of the residuals, the autocorrelation of

the residual and thep- values of the Ljung-Box statistic for the first 10 lags.

The functionBox.test()computes the test statistic for a given lag

Box-test(fit$residuals, lag=1)

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Prediction of ARMA-Models

predict( )can be used for predicting future values of the levels under themodel

AR.pred


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1 INTRODUCTION

2 ESTIMATION OF


3 ASYMPTOTIC

PROPERTIES OF MLESTIMATORS AND

INFERENCE


ARMA PROCESS:

SELECTINGp ANDq


6 FORECASTING


IN THE GENERAL CASE

8 R CODE

9 REFERENCES

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Brockwell and Davies, (1991), Chapters 8, 9


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Hamilton, 1994, Chapter 5

Wei, Chapters 6,7.

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ARMA Estimation