# BUS 10123 Chapter Notes - Chapter 6: Autoregressive Model, Partial Autocorrelation Function, AutocorrelationExam

by OC2511532

School

Kent State UniversityDepartment

Business Administration InterdisciplinaryCourse Code

BUS 10123Professor

Eric Von HendrixStudy Guide

FinalThis

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1

1. Autoregressive models specify the current value of a series yt as a function of its

previous p values and the current value an error term, ut, while moving average

models specify the current value of a series yt as a function of the current and

previous q values of an error term, ut. AR and MA models have different

characteristics in terms of the length of their “memories”, which has implications for

the time it takes shocks to yt to die away, and for the shapes of their autocorrelation

and partial autocorrelation functions.

2. ARMA models are of particular use for financial series due to their flexibility. They

are fairly simple to estimate, can often produce reasonable forecasts, and most

importantly, they require no knowledge of any structural variables that might be

required for more “traditional” econometric analysis. When the data are available at

high frequencies, we can still use ARMA models while exogenous “explanatory”

variables (e.g. macroeconomic variables, accounting ratios) may be unobservable at

any more than monthly intervals at best.

3. yt = yt-1 + ut (1)

yt = 0.5 yt-1 + ut (2)

yt = 0.8 ut-1 + ut (3)

(a) The first two models are roughly speaking AR(1) models, while the last is an

MA(1). Strictly, since the first model is a random walk, it should be called an

ARIMA(0,1,0) model, but it could still be viewed as a special case of an

autoregressive model.

(b) We know that the theoretical acf of an MA(q) process will be zero after q lags, so

the acf of the MA(1) will be zero at all lags after one. For an autoregressive process,

the acf dies away gradually. It will die away fairly quickly for case (2), with each

successive autocorrelation coefficient taking on a value equal to half that of the

previous lag. For the first case, however, the acf will never die away, and in theory

will always take on a value of one, whatever the lag.

Turning now to the pacf, the pacf for the first two models would have a large

positive spike at lag 1, and no statistically significant pacf’s at other lags. Again, the

unit root process of (1) would have a pacf the same as that of a stationary AR

process. The pacf for (3), the MA(1), will decline geometrically.

(c) Clearly the first equation (the random walk) is more likely to represent stock

prices in practice. The discounted dividend model of share prices states that the

current value of a share will be simply the discounted sum of all expected future

dividends. If we assume that investors form their expectations about dividend

payments rationally, then the current share price should embody all information that

is known about the future of dividend payments, and hence today’s price should

only differ from yesterday’s by the amount of unexpected news which influences

dividend payments.

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Thus stock prices should follow a random walk. Note that we could apply a similar

rational expectations and random walk model to many other kinds of financial series.

If the stock market really followed the process described by equations (2) or (3), then

we could potentially make useful forecasts of the series using our model. In the

latter case of the MA(1), we could only make one-step ahead forecasts since the

“memory” of the model is only that length. In the case of equation (2), we could

potentially make a lot of money by forming multiple step ahead forecasts and

trading on the basis of these.

Hence after a period, it is likely that other investors would spot this potential

opportunity and hence the model would no longer be a useful description of the

data.

(d) See the book for the algebra. This part of the question is really an extension of

the others. Analysing the simplest case first, the MA(1), the “memory” of the

process will only be one period, and therefore a given shock or “innovation”, ut, will

only persist in the series (i.e. be reflected in yt) for one period. After that, the effect

of a given shock would have completely worked through.

For the case of the AR(1) given in equation (2), a given shock, ut, will persist

indefinitely and will therefore influence the properties of yt for ever, but its effect

upon yt will diminish exponentially as time goes on.

In the first case, the series yt could be written as an infinite sum of past shocks, and

therefore the effect of a given shock will persist indefinitely, and its effect will not

diminish over time.

4. (a) Box and Jenkins were the first to consider ARMA modelling in this logical

and coherent fashion. Their methodology consists of 3 steps:

Identification - determining the appropriate order of the model using

graphical procedures (e.g. plots of autocorrelation functions).

Estimation - of the parameters of the model of size given in the first stage.

This can be done using least squares or maximum likelihood, depending on

the model.

Diagnostic checking - this step is to ensure that the model actually estimated

is “adequate”. B & J suggest two methods for achieving this:

- Overfitting, which involves deliberately fitting a model larger than that

suggested in step 1 and testing the hypothesis that all the additional

coefficients can jointly be set to zero.

- Residual diagnostics. If the model estimated is a good description of the

data, there should be no further linear dependence in the residuals of the

estimated model. Therefore, we could calculate the residuals from the

estimated model, and use the Ljung-Box test on them, or calculate their acf. If

either of these reveal evidence of additional structure, then we assume that

the estimated model is not an adequate description of the data.

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

If the model appears to be adequate, then it can be used for policy analysis

and for constructing forecasts. If it is not adequate, then we must go back to

stage 1 and start again!

(b) The main problem with the B & J methodology is the inexactness of the

identification stage. Autocorrelation functions and partial autocorrelations

for actual data are very difficult to interpret accurately, rendering the whole

procedure often little more than educated guesswork. A further problem

concerns the diagnostic checking stage, which will only indicate when the

proposed model is “too small” and would not inform on when the model

proposed is “too large”.

(c) We could use Akaike’s or Schwarz’s Bayesian information criteria. Our

objective would then be to fit the model order that minimises these.

We can calculate the value of Akaike’s (AIC) and Schwarz’s (SBIC) Bayesian

information criteria using the following respective formulae

AIC = ln ( ) + 2k/T

SBIC = ln ( ) + k ln(T)/T

The information criteria trade off an increase in the number of parameters

and therefore an increase in the penalty term against a fall in the RSS,

implying a closer fit of the model to the data.

5. The best way to check for stationarity is to express the model as a lag polynomial

in yt.

y y y u

t t t t

= + +

0 803 0 682

1 2

. .

Rewrite this as

y L L u

t t

( . . )1 0 803 0 682 2

=

We want to find the roots of the lag polynomial

( . . )1 0 803 0 682 0

2

=L L

and

determine whether they are greater than one in absolute value. It is easier (in my

opinion) to rewrite this formula (by multiplying through by -1/0.682, using z for the

characteristic equation and rearranging) as

z2 + 1.177 z - 1.466 = 0

Using the standard formula for obtaining the roots of a quadratic equation,

= 0.758 or 1.934

Since ALL the roots must be greater than one for the model to be stationary, we

conclude that the estimated model is not stationary in this case.

6. Using the formulae above, we end up with the following values for each criterion

and for each model order (with an asterisk denoting the smallest value of the

information criterion in each case).

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