Time Series Analysis. PART II. Econometric Forecasting
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Time Series Analysis
APPROACHES TO ECONOMIC FORECASTING
(2) single-equation regression models
(3) simultaneous-equation regression models
(4) Autoregressive integrated moving average models (ARIMA)
(5) vector autoregression.
ARIMA methodology, the emphasis of these methods is not on constructing single-equation or simultaneous-equation models but on analyzing the probabilistic, or stochastic, properties of economic time series on their own under the philosophy let the data speak for themselves.
Unlike the regression models, in which Ytis explained by k regressor X1, X2, X3, . . . , Xk, the BJ-type time series models allow Yt to be explained by past, or lagged, values of Y itself and stochastic error terms.
Autoregressive processes (AR)
An AR(1) process is written as:
yt = yt-1 + t where t ~ IID(0,2)
ie. the current value of yt is equal to times its previous value plus an unpredictable component t .we say that Yt follows a first-order autoregressive, or AR(1), stochastic process.
Here the value of Y at time t depends on its value in the previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (= ) of its value at time (t − 1) plus a random shock or disturbance at time t.
If we consider a model
yt = 1yt-1 + 2yt-2 + t
then we say that Yt follows a second-order autoregressive, or AR(2), process. That is, the value of Y at time t depends on its value in the previous two time periods.
This can be extended to an AR(p) process
yt = 1yt-1 + 2yt-2 +…..+pyt-p + t
where t ~ IID(0,2)
ie. the current value of yt depends on p past values plus an unpredictable component t .
Moving Average processes(MA)
An MA(1) process is written as
yt = t + β1t-1where t ~ IID(0,2)
ie. the current value is given by an unpredictable component t and β times the previous period’s error.
But if Y follows the expression,
yt = t + β1t-1 + β2t-2
then it is an MA(2) process,
This can also be extended to an MA(q) process
yt = t + β 1t-1 +…..+ β qt-q
where t ~ IID(0,2)
In short, a moving average process is simply a linear combination of white noise error terms.
Autoregressive and Moving Average (ARMA) Process
Of course, it is quite likely that Y has characteristics of both AR and MA and is therefore ARMA. Thus,Yt follows an ARMA(1, 1) process if it can be written as
yt = 1yt-1 + t + β1t-1
Because there is one autoregressive and one moving average term.
In general, in an ARMA( p, q) process, there will be p autoregressive and q moving average terms,
yt = 1yt-1 +...+ pyt-p +t + β1t-1 +...+ βqt-q
if a time series is integrated of order 1 [i.e., it is I(1)], its first differences are I(0), that is, stationary. Similarly, if a time series is I(2), its second difference is I(0). In general, if a time series is I(d), after differencing it d times we obtain an I(0) series. Therefore, if we have to difference a time series d times to make it stationary and then apply the ARMA(p, q) model to it,
we say that the original time series is ARIMA(p, d, q), that is, it is an autoregressive integrated movingaverage time series, where p denotes the number of autoregressive terms, d the number of times the series has to be differenced before it becomes stationary, and q the number of moving average terms.
Thus, an ARIMA(2, 1, 2) time series has to be differenced once (d = 1) before it becomes stationary and the (first-differenced) stationary time series can be modeled as an ARMA(2, 2) process, that is, it has two AR and two MA terms.
the values of p and q) or an ARIMA process, in which case we must know the values of p, d, and q. The BJ methodology comes in handy in answering the preceding question. The method consists of four steps:
To find out the appropriate values of p, d,
and q. We use the correlogram and partial correlogram for this task.
Having identified the appropriate p and q values, the next stage is to estimate the parameters of the autoregressive and moving
average terms included in the model. Sometimes this calculation can be done by simple least squares but sometimes we will have to resort to nonlinear (in parameter) estimation methods.
each case must be checked.
The term autoregressive is due to the appearance of the lagged value of the
dependent variable on the right-hand side and the term vector is due to the
fact that we are dealing with a vector of two (or more) variables.
Financial time series, such as stock prices, exchange rates, inflation rates, etc. often exhibit the phenomenon of volatility clustering, that is, periods in which their prices show wide swings for an extended time period followed by periods in which there is
Knowledge of volatility is of crucial importance in many areas. For example, considerable macroeconometric work has been done in studying the variability of inflation over time. For some decision makers, inflation in itself may not be bad, but its variability is bad because it makes financial planning difficult.
On the other hand, in the first difference form, they are generally stationary
Therefore, instead of modeling the levels of financial time series, why not model their first differences? But these first differences often exhibit wide swings, or volatility, suggesting that the variance of financial time series varies over time.
How can we model such “varying variance”? This is where the so-called autoregressive conditional heteroscedasticity (ARCH)