Sunday, January 16, 2011

Forecasting and Time Series Analysis


Time Series Analysis

Exponential Smoothing:

S = α Y t +  (1-α) Yt-1              
S =  portion of current average + portion of previous average = smooth value

Classical Equation:

Forecasted Value = T x S x C                        
FV =trend*seasonal index*cyclical index} 

Regression Equation:

T + S + C    
Y = trend regression with seasonal and cyclical dummies

Seasonal Indices: 

AVGseason i  /  AVGtotal  = seasonal index for season i.


Econometric Forecasting Models

Distributed Lag Models

Yt = a0 + b0 X t +  λYt-1  +  ut

where λ is on the interval (0,1)  
 
et = r et-1  +  ut    (serially correlated error terms)

Serially correlated error terms can be corrected using GLS, iterated maximum likelihood, instrumental variables.

Non-stationary: upward or downward trend in data, this can be corrected by take first differences:
           
         Y* = Y­t – Yt-1

Application: To predict  Yt+1 use the current value of Y.  Predicting Yt+2 can be predicted as a conditional forecast, given a forecast of Yt+1 

ARIMA Models

Combines moving average (MA) and autoregressive (AR) models as well as differencing.

ARIMA(p,d,q)  where p = # of autoregressive terms
                                    d = # of 1st differences
                                    q = # moving average terms

Application:  ARIMA(2,1,1) :    Y*t = b0 + b1Y*t-1  + b2Y*t-2  + b3 et-1  + et

2 autoregressive terms,  1 1st difference, 1 moving average term

Seasonal effects can also be incorporated:

ARIMA(p,d,q)(P,D,Q)


where P = # of seasonally differenced autoregressive terms
          D = # of seasonal differences
          Q = # of seasonally differenced moving average terms


This model can also be represented as such:

[θ(B) θ(Bs) / Φ(B) Φ(Bs) ΔdΔs  ] *   a
where  
B = backshift operater such that BY = Yt-1
θ(B) = MA(q)
θ(Bs) = seasonal MA(Q)
Φ(B) = AR(p)
Φ(Bs) = seasonal AR(P)
Δd=  differencing
Δs = seasonal differencing
at = white noise term

References:

Using Econometrics: A Practical Guide (4th Edition)
A.H. Studenmund

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