This is a model with three equations, and with three en- dogenous … · 2018. 9. 6. · Notes From ‚Applied Econometric Time Series, Walter Enders™ Some De–nitions Stochastic

Notes From Applied Econometric Time Series, Walter Enders

Some Denitions

Stochastic version of Samuelsons (1939) Classical Model

yt = ct + it

ct = �yt�1 + "ct

it = �(ct � ct�1) + "it� This is a model with three equations, and with three en-dogenous variables (yt; ct; it)

� It is a dynamic model (Past variables a¤ect the currentvariables)

1


� It is a structural form model. This is because it explainsendogenous variables with current realizations of other en-dogenous variables

A reduced form model explains endogenous variableswith exogenous ones

� Note that the lags of endogenous variables are alsoexogenous to the model

2


� Reduced form investment equation can be obtained as

it = (ct � ct�1) + �it = (�yt�1 + "ct � ct�1) + "it= �yt�1 � ct�1 + "ct + "it;

which can also be written as

it = �1yt�1 + �2ct�1 + et

3


� Similarly, after some substitutions a reduced-form equa-tion for GDP can be obtained as follows

yt = ayt�1 + byt�2 + et

� This is a univariate reduced-form equation; yt is expressedsolely as a function of its own lags and a disturbance term

4


CHAPTER 1: DIFFERENCE EQUATIONS

� Di¤erence equation expresses the value of a variable as afunction of its own lagged values, time, and other variables

� Time-series econometrics is concerned with the estimationof di¤erence equations containing stochastic components

5


� Suppose we have the following series as data

6


� Time series methodology was originally developed to de-compose a series into a trend, a seasonal,a cyclical, and anirregular components

The trend component represents the long-term behav-ior of the series

The cyclical and seasonal components represent the reg-ular periodic movements

The irregular component is stochastic

7


Di¤erence Equations and Their Solutions

� n th-order di¤erence equation with constant coe¢ cients

yt = a0 +

nXi=1

aiyt�i + xt (10)

where xt =1Xi=0

�i"t�i

� The question is this equation is stable or not?

8


� Consider the rst order homogeneous di¤erence equation

yt = a0 + a1yt�1 + "t (17)

� If ja1j < 1, the series eventually uctuates around a con-stant number

9


E-views Application

wfcreate (wf=income process) u 50series y=5

smpl @rst+1 @lasty=0.5*y(-1)+2

smpl @allgraph aa yshow aa

10


3.8

4.0

4.2

4.4

4.6

4.8

5.0

5.2

5 10 15 20 25 30 35 40 45 50

Y

11


� If j a1 j> 1, then the fytg series explodes

E-views Application


smpl @rst+1 @lasty=1.5*y(-1)+2


12


0

500,000,000

1,000,000,000

1,500,000,000

2,000,000,000

2,500,000,000

3,000,000,000

3,500,000,000

4,000,000,000

5 10 15 20 25 30 35 40 45 50

Y

13


� If a1 = 1, the di¤erence quation can be written as

yt = a0t +tXi=0

"i + y0

tXi=1

"i is the random walk component

a0t is the time trendtogether, the fytg series follow a random walk with adrift

14


E-views Application (Drift Component-also called Trend Com-ponent)wfcreate (wf=income process) u 500series y=5smpl @rst+1 @lasty=y(-1)+2


15


0

200

400

600

800

1,000

1,200

50 100 150 200 250 300 350 400 450 500

Y

16


E-views Application (Unit Root Component)wfcreate (wf=income process) u 50series y=5series e=nrnd

smpl @rst+1 @lasty=y(-1)+e


17


-25

-20

-15

-10

-5

0

5

10

50 100 150 200 250 300 350 400 450 500

Y

18


Solving Higher Order Homogeneous Di¤erence Equations

� Consider the following second order homogeneous di¤er-ence equations

yt = 0:2yt�1 + 0:35yt�2

� As you might guess, determining the stability conditionsfor this equation depends knowledge on more than oneparameters

� See Applied Econometric Time Series (Walter Enders) tocheck necessary and su¢ cient conditions for stability ofhigher order systems

19


CHAPTER 2: STATIONARY TIME-SERIES MODELS

Stochastic Di¤erence Equation Models

� Say we have sequence of observations of a variable y overtime

fy0; y1; y2; : : : ; ytg� A stochastic process describes the probability structure ofthese observations

� The elements of an observed time series are realizations ofthis stochastic process

20


Stationarity

� A stochastic process is called weakly (covariance) station-ary when the mean, the variance and the covariance struc-ture of the process is time independent and nite, thatis

E(yt) = �


A White-Noise Process

� It is the simplest stationary process� A sequence is a white-noise process if it has a mean ofzero, a constant variance, and is uncorrelatedwith all otherrealizations. Formally,

E("t) = 0

V ar("t) = 0 = �2"

E("t"t�j) = j = 0 for j = 1; 2; :::

22


ARMA Models

� For a stochastic process, if stability conditions hold, sta-tionarity conditions are satised

�Wolds Decomposition Theorem: Any discrete stationarycovariance time series process fytg can be expressed as thesum of two uncorrelated processes

yt = dt + ut

where dt is purely deterministic and ut is a purely indeter-ministic process:

ut =1Xi=0

�i"t�i

whereP1

i=0(�i)2 < 1 is necessary for stationarity (it is

23


conventional to dene �0 = 1) , and "t is a white noiseprocess

� Taking dt as a constant, reparametrizing the indetermin-istic process (below we discuss the way doing it)

yt = a0 +

pXi=1

aiyt�i +

qXi=0

�i"t�i: (5)

If all the characteristic roots of (5) are all in the unit circle,fytg is called an ARMA model for fytg

24


� The autoregressive part:pXi=1

aiyt�i

� The moving average part:qXi=0

�i"t�i

� If the homogeneous part of the di¤erence equation containsp lags and the moving average part contains q lags, themodel is called an ARMA(p; q) model

25


Why use ARMA Models?

� Suppose the true data generating process is as follows

yt = c + xt + "t + "t�1

where xt exogenous regressors and "t is a white noise process

� The above process suggests that shocks to yt lasts for twoperiods

� If you estimate the above model through a regression

yt = c + xt + et

then there is a serial correlation in the error terms,

cov(et; et�1) = cov("t + "t�1; "t�1 + "t�2) = var("t�1) 6= 0

26


� Serial correlation violates the standard assumption of re-gression theory that error terms are uncorrelated

Reported standard errors and t-statistics are invalid

� Things can even get worse: Suppose the true data gener-ating process is as follows

yt = c + yt�1 + xt + "t + "t�1

and if you estimate it by using the following model

yt = c + yt�1 + xt + et

regressors and the error terms become correlated

cov(yt�1; et) = cov(yt�1; "t + "t�1) 6= 0In this case, OLS estimates are biased and inconsistent

27


E-views Application


series e=nrnd

smpl @rst+1 @lasty=0.7*y(-1)+e+0.5*e(-1)

ls y y(-1)

28


29


� Serial correlation is a common occurrence in time seriesdata

� If accounted for, this time dependence is useful. It allowus to understand and predicting future values of series

� ARMA models accounts for this time dependence so thatthe model captures all of the relevant structure

30


� In what follows:1. We will use stationary data� So that the mean, variance, and autocorrelations ofthe series can be obtained based on the single set ofrealizations

2. Given the stationary data, we will identify the datagenerating process (type of ARMA model)� For this, we will use autocorrelation and partial au-tocorrelation functions

� There are methods to make the nonstationary data sta-tionary, such as di¤erencing, detrending, and ltering. Wedefer this discussion to the next chapters

31


The Autocorrelation (ACF) and Partial Autocorrelation (PACF)Functions

� In the case of stationary processes, the autocorrelation co-e¢ cient at lag j, denoted by �j, is dened as the correla-tion between yt and yt�j:

�j =cov(yt; yt�s)p

var(yt)pvar(yt�j)

=

j

0; j = 0;�1;�2; :::

� The plot of �j against j (for j � 1) is called correlogram

The properties of autocorrelation function (ACF) are:

�0 = 1

j�jj � 1

32


� The partial autocorrelation coe¢ cient, on the other hand,measures the linear association between yt and yt�j ad-justed for the e¤ects of the intermediate values yt�1; :::; yt�j+1

� Therefore, it is the coe¢ cient aj in the linear regressionmodel:

yt = a0 + a1yt�1 + ::: + ajyt�j + et

33


�Wewill examine three extreme cases forARMA(p; q)model

yt = a0 +

pXi=1

aiyt�i +

qXi=0

�i"t�i (5)

1. White Noise Process (a0 = p = q = 0)

2. Pure Autoregressive Process (q = 0)

3. Pure Moving Average Process (p = 0)

34


1- White Noise Process

� The simplest ARMA model is a white noise process

yt = "t

which has no memory

�We can easily show that this is a stationary process

E(yt) = 0


� Autocovariance function of yt is:

j =

��2" j = 00 j 6= 1

�� Its autocorrelation (ACF) and partial autocorrelation (PACF)functions are:

�j =

�1 j = 00 j 6= 1

�aj =

�1 j = 00 j 6= 1

�

36


E-views Application

wfcreate (wf=income process) u 5000series e=nrndseries y=egraph aa yshow aa

37


y.correl(10)

Hence, a white noise process in an ARMA(0,0) model; itshows no history dependence in any form

38


2- Pure Autoregressive Process

� Example: AR(1) Processyt = a0 + a1yt�1 + "t

(1� a1L)yt = a0 + "tyt =

a01� a1L

+"t

1� a1L� If j a1 j< 1, the last equation can be written as

yt =a0

1� a1+

1Xi=0

ai1"t�i

� AR process has an innite memory so that it can be writ-ten as a collection of past shocks

39


� Notice that yt is a stationary process� Its autocorrelation (ACF) and partial autocorrelation (PACF)functions are as follows:

�j =

j

0= aj1 aj =

�a1 j = 10 j > 1

�� Thus, ACF converges to zero geometrically� PACF is zero at lag 2 and greater

40


E-views Application

wfcreate (wf=income process) u 300series e=nrndseries y=0smpl @rst+1 @lasty=0.5*y(-1)+egraph aa yshow aa

41


-3

-2

-1

0

1

2

3

4

25 50 75 100 125 150 175 200 225 250 275 300

Y

42


y.correl(10)

43


ls y c ar(1)series res=residres.correl(10)

44


� Summary:� Autoregressive processes have an exponentially decliningACF, whether they are AR(1), AR(2), etc.

� The partial autocorrelation of an AR(p) process is zero atlag p+1 and greater

� Note: Nonstationary series also have an ACF that remainssignicant for some lags rather than quickly declining tozero

45


3- Pure Moving Average Process

yt = a0 +

qXi=0

�i"t�i = a0 + �(L)"t

� Moving average processes have a geometrically (or oscil-latory) declining PACF; hence, they cannot be used inidentifying the order of an autoregressive model

� Hence, the autocorrelation of an MA(q) process is zero atlag q+1 and greater

46


E-views Application

wfcreate (wf=income process) u 300series e=nrnd/15series y=0smpl @rst+1 @lasty=e+0.5*e(-1)graph aa yshow aa

47


-.2

-.1

.0

.1

.2

.3

25 50 75 100 125 150 175 200 225 250 275 300

Y

48


y.correl(10)

49


ls y c ma(1)series res=residres.correl(10)

50


Alternative Methods of Checking for Serial Correlation

� The last two columns reported in the correlogram are theLjung-Box Q-statistics and their p-values. The Q-statisticat lag is a test statistic for the null hypothesis that thereis no autocorrelation up to order and is computed as

Q = TsXk=1

r2k

high sample autocorrelations lead to large values of Q

51


Parsimony

� Incorporating additional coe¢ cients to an ARMA modelwill necessarily increase t of the model at a cost of reduc-ing degrees of freedom

� A parsimonious model ts the data well without incorpo-rating any needless coe¢ cients

� Box and Jenkins argue that parsimonious models producebetter forecasts than overparameterized models

� If di¤erent ARMA models may have similar properties,such as AR(1) and MA(1), the AR(1) model is the moreparsimonious model and is preferred

52


Model Selection Criteria

�We never know the true data-generating process� There exist various model selection criteria that trade-o¤a reduction in the sum of squares of the residuals for amore parsimonious model

� The two most commonly used model selection criteria arethe Akaike Information Criterion (AIC) and the SchwartzBayesian Criterion (SBC)

AIC = T � ln(sum of squared residuals) + 2nSBC = T � ln(sum of squared residuals) + n � ln(T )where n = number of parameters estimated (p + q +possible constant term)

53


T = number of usable observations (some observationsare lost with a model using lagged variables)

� Notice that increasing the number of regressors increasesn but reduces the sum of squared residuals (SSR)

� Ideally, the AIC and SBC will be as small as possible� EViews and SAS report values for the AIC and SBC using

AIC� = �2ln(L)=T + 2n=TSBC� = �2ln(L)=T + nln(T )=T

where L is the maximum of the log of the likelihood func-tion

� For a normal distribution,�2ln(L) = T ln(2�)+T ln(�2)+(1=�2)(SSR)

54


CHAPTER 3: MODELS WITH TRENDS (NONSTATION-ARY MODELS)

� Deterministic Trends (Drift Model)

yt = yt�1 + a0 (or �yt = a0)

� Stochastic Trend (Random Walk Model)

yt = yt�1 + "t (or �yt = "t)

� The Random Walk Plus Drift Model

yt = yt�1 + a0 + "t (or �yt = a0 + "t)

� All series are nonstationary

55


REMOVING THE TREND

� A reason for trying to stationarize a time series is to beable to obtain meaningful sample statistics such as means,variances, and correlations with other variables

� Moreover, using non-stationary time series data producesunreliable and spurious results and leads to poor under-standing and forecasting

� Below, we analyze methods for making a series stationary,appropriateness of which depend on whether the trend hasa deterministic, or a stochastic component

56


Di¤erencing

� First consider the solution for the random walk plus driftmodel

yt = yt�1 + a0 + "tTaking the rst di¤erence, we obtain

�yt = a0 + "t (1)

Clearly, the �yt sequence equal to a constant plus awhite-noise disturbance is stationary

� The above process is calledARIMA(0,1,0)model (ARIMA:autoregressive integrated moving average)

� The model has no AR and MA component, but requiresrst di¤erencing to be stationary

57


� Di¤erencing the data eliminates most of the serial correla-tion and transforms an I(1) process to an I(0)

� The general point is that the d th di¤erence of a processwith d unit roots is stationary

� These models are called Di¤erence Stationary Models� Such a sequence is integrated of order d and denoted byI(d)

58


Detrending

� Consider, for example, a model that is the sum of a deter-ministic trend and a pure noise component:

yt = y0 + a1t + "t

� An appropriate way to transform this model is to estimatethe regression

yt = a0 + a1t + et;

where et is the estimated values of the "t series, and a0 isthe estimated value of the y0

� Simply subtracting the estimated values of the yt sequencefrom the actual values (detrending) yields an estimate ofthe stationary sequence et

59


� Hence, the above model is called Trend Stationary Model� The detrended (or di¤erenced) processes can then be mod-eled using traditional methods (such as ARMAestimation)

� In general, detrending is accomplished by regressing yt ona deterministic polynomial time trend

yt = a0 + a1t + a2t2 + ::: + ant

n + et

60


The E¤ect of a Unit Root on Regression Residuals

� Consider the regression equationyt = a0 + a1zt + et (4.12)

� The assumptions of the classical regression model necessi-tate that both the yt and zt sequences be stationary andthat the errors have a zero mean and a nite variance

� In the presence of nonstationary variables, there might bewhat Granger and Newbold (1974) call a spurious regres-sion

� A spurious regression has a high R2 and t-statistics thatappear to be signicant, but the results are without anyeconomic meaning.

61


� The regression output looks good,but the least-squaresestimates are not consistent and the customary tests ofstatistical inference do not hold

� Lets generate two sequences,yt and zt, as independent ran-dom walks using the formulas:

yt = 0:2 + yt�1 + "yt

zt = �0:1 + zt�1 + "ztwhere "yt and "zt are white-noise processes that are inde-pendent of each other

� Since the yt and zt sequences are independent of eachother, (4.12) is necessarily meaningless; any relationshipbetween the two variables is spurious

62


� Granger and Newbold (1974), on the other hand, were ableto reject the null hypothesis a1 = 0 in approximately 75%of the cases

The main reason is that although it is the deterministicdrift terms that cause the sustained increase in yt andthe overall decline in zt, it appears that the two seriesare inversely related to each other and the correlationbetween the variables gets close to 1

63


� The Results:� If the yt and zt sequences are integrated of di¤erent orders,regression equations using such variables are meaningless

� If the nonstationary yt and zt sequences are integrated ofthe same order, and if the residual sequence contains astochastic trend, the regression is spurious

In this case, it is often recommended that the regressionequation be estimated in rst di¤erences

�yt = a1�zt +�et

� If the nonstationary yt and zt sequences are integrated ofthe same order, and if the residual sequence is stationary,yt and zt are cointegrated

64


Dickey-Fuller Tests

� This section outlines a procedure to determine whethera1 = 1 in the model

yt = a1yt�1 + et

� Begin by subtracting yt�1 from each side of the equationin order to write the equivalent form

�yt = yt�1 + "t

where = a1 � 1� Testing the hypothesis a1 = 1 is equivalent to testing thehypothesis = 0

65


� Dickey and Fuller (1979) consider three di¤erent regressionequations that can be used to test for the presence of a unitroot:

�yt = yt�1 + "t

�yt = a0 + yt�1 + "t

�yt = a0 + yt�1 + a2t + "t

� The parameter of interest in all the regression equations is

; if = 0, the yt sequence contains a unit root

� Imposing the constraints a0 = 0 and a2 = 0 correspondsto a pure random walk model, if a0 6= 0 a drift term isadded, if a2 6= 0 a time trend is added

66


� The critical values of each test are unchanged when thelast equations above are replaced by the autoregressiveprocesses:

�yt = yt�1 +

pXi=2

�i�yt�i+1 + "t

�yt = a0 + yt�1 +

pXi=2

�i�yt�i+1 + "t

�yt = a0 + yt�1 + a2t +

pXi=2

�i�yt�i+1 + "t

� The thing is we cannot properly estimate and its stan-dard error unless all of the autoregressive terms are in-cluded in the estimating equation

67


� Tests including lagged changes are calledAugmented DickeyFuller tests

� Since the true order of the autoregressive process is un-known, we need to select the appropriate lag length

68


Selection of the Lag Length

� Too few lags mean that the regression residuals do notbehave like white-noise processes

� Including too many lags reduces the power of the test anda loss of degrees of freedom

� One approach to select the appropriate lag length is thegeneral-to-specic methodology. The idea is to start witha relatively long lag length and pare down the model bythe usual t-test and/or F-tests, or by AIC an SBC tests

� Done automatically by E-views

69


Example: The le HW2_data.xlsx contains the U.S. datafrom column F onwards. Use the real GDP data in the le

a-) Form the log of real GDP as lyt = log(RGDP ). Formthe autocorrelations. By using augmented DickeyFuller testcheck if the series is stationary

b-) Form the log of real GDP as lyt = log(RGDP ). Detrendthe data with a linear time trend and obtain the residuals.Then check if the residual series is stationary

c-) Find the growth rayte of real GDP as lyt � lyt�1. Thencheck if the series is stationary.

d-) Using ACF and PACF, and also the Akaike InformationCriterion (AIC) and the Schwartz Bayesian Criterion (SBC)model selection criterias, nd the parsimonous model that

70


best represent the growth rate of real GDP data (nd theARMA representation of the data). Compare your result withthe lag length selected automatically in the unit root test inpart c-).

71

Documents

This is a model with three equations, and with three en- dogenous … · 2018. 9. 6. · Notes From ‚Applied Econometric Time Series, Walter Enders™ Some De–nitions Stochastic