Time series ARIMA model

 Time series and some specific model

I am going to explain how to analyse the behaviour and variability of dependent variables by regressions it using a number of different regressors or explanatory variables.

there are various aspects to time series analysis but one common theme to them all is full used of dynamic structure of the data bye by this we mean that we extract as much informations as possible from the past history of the series. The two principles types of time series analysis are time series forecasting and dynamic modelling.

Time series forecasting the ARIMA model

AR=autoregressive

I=integrated

MA=moving average

we will begin by examining the simplest model the autoregressive model of order 1 then continue with the survey of ARIMA a model finally the box Jenkins approach for a model selection and forecasting will be presented.

Stationarity

Concept underlines time series process is that of stationery. Time series is covariance stationery when it has three characteristics,

E(Yt)=constant for all t

VAR(Yt)=constant for all t

Cov(Yt,Yt+k)=constant for all t and all k will not equal to zero.

If the series is non-stationary all the typical results of the classical regression analysis are not valid

Regression with non-stationary series may have no meaning and there are called a spurious. Socks to a stationary time series are necessary temporary over time, the effect of the shocks will display and series will reward to its long-run mean level.

Autoregressive time series models

Autoregressive of order one model AR(1)

Yt=phiYt-₁₊ᵤt

We do not include a constant and modulus phi less than 1 and ut is a Gaussian error term. AR(1) model is that the time-series behaviour of Yt is largely determined by its own value in the preceding period.

Condition for stationarity

The constant mode phi less than 1 in order to guarantee stationarity as defined in the previous section.

if we have modulus phi greater than 1 then Yt will tend to get larger in each period so we would have to explosive series.

Moving average model

The general form of MA(1) model

Yt=ut+theta(ut-1)

Yt depends on the value of the immediate past error which is known at time t.

Box Jenkins model

Box and Jenkins popular 3 stage method aimed at selecting and appreciate an ARIMA model for the purposes of estimating and forecasting of univariate time series. There are three stages,

Identification

Estimations

Diagnostic checking.

Identification; the researcher visually examine the time plot of the series ACF and PACF, if the series is non-stationary the ACF of the series will not die down or so sign of decay at all. If this is the case the series needs to be transformed to make it stationary.if neither the acf nor the PAC show definite cut off a mixed process is suggested. In this case it is a difficult but not impossible to identify the AR and MA order we should think the acf and pacf of pure ARIMA process as being shoe superimposed onto one another.

Ljung–Box test. 

 Box.test(diff(y), lag = 10, type = "Ljung-Box")

 Box.test(ret, lag = 5, type = "Ljung-Box")

	Box-Ljung test

data:  ret
X-squared = 53.87, df = 5, p-value
= 2.229e-10