Time Series Analysis in Econometrics: AR, MA, ARMA, and ARIMA Models

Actual- fitted and residual
graph – making a residual
series – Assumptions of
Classical Linear Regression
Model-AR-MA-ARMA-ARIMA

In the context of regression analysis or model
fitting, the actual vs fitted and residual
graphs are used to evaluate the performance
of a model.

Actual vs Fitted Graph
• This graph compares the observed (actual) values from the dataset
with the predicted (fitted) values generated by the model.
• On the x-axis, we plot the actual values, and on the y-axis, we plot the
fitted (predicted) values.
• Ideally, the points should lie along a 45-degree line (the line of perfect
prediction), showing that the model has correctly predicted the
values.

Residual Graph
• A residual is the difference between the actual value and the fitted
(predicted) value:
Residual=Actual−Fitted
• A residual plot displays the residuals on the y-axis and the fitted
values on the x-axis (or sometimes the actual values).
• This plot helps to assess whether there are patterns in the residuals
that indicate issues with the model. Ideally, residuals should be
randomly scattered around zero without any discernible patterns.

Assumptions of Classical Linear
Regression Model
• Linear Model
• No Multicollinearity
• No Autocorrelation
• No Heteroscedasticity
• No specification Bias
• Mean if Error term [E(u)=0]
• Variance of the error term in
• X values are fixed in repeated sampling
• Covariance between Xi and Ui is Zero
• The number of observation “n” must be greater than or equal to the number of
parameters to be estimated.

Why AR/MA/ARMA/ARIMA
AR---->Many real-world time series (stock prices, economic indicators, etc.)
depend on their past values.AR models help identify and use this relationship
for forecasting.
MA--Some time series contain random shocks (unexpected variations). MA
models smooth out these fluctuations.
ARMA Some time series require both past values and past errors for
effective modeling.
ARIMA- Handles Non-Stationary Data. Real-world time series often have
trends and seasonal patterns. AR and MA require stationarity, but ARIMA
includes a differencing step (I for Integration) to handle trends. ARIMA can
model almost any real-world time series after proper transformation.

AR,MA and ARIMA Modelling of TSD

An Autoregressive (AR) Process
• Let Yt represents the logged GDP at time t.
• Model Yt :
where δ is the mean of Y and where ut is an uncorrelated random error
term with zero mean and constant variance σ2 (i.e., it is white noise),
then we say that Yt follows a first-order autoregressive, or AR(1),
stochastic process.

Second order Autoregressive model
[AR(2)]

Moving Average /MA
• Suppose we model Yt as

ARMA model(Autoregressive and Moving
Average )

Autoregressive Integrated Moving
Average(ARIMA)-Box-junkins (JB) Methodology
• If we have to difference a time series ‘d’times to make it stationary
and then apply ARMA(p,q) model to it , we say that the original time
series is ARIMA(p,d,q),ie.,ARIMA model
• Where p =number of AR terms q=Number of MA terms and
d=number of times the series has to be differences to make it
stationary
• Eg: ARIMA(2,1,2) means the TS has been differenced once before it
becomes stationary and the first differences stationary TS can be
modelled as ARMA(2,2), which means it hgas 2 AR terms and 2 MA
terms .

• ARIMA(p,d=0,q)=?
• ARIMA(p,0,0)=?
• ARIMA(0,0,q)
Stationary or non stationary

Time Series Analysis in Econometrics: AR, MA, ARMA, and ARIMA Models

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