Forecasting and Time Series Analysis

Time Series Analysis

Exponential Smoothing:

S = α Y _t +  (1-α) Y_t-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: 

AVG_{season i}  /  AVG_total  = seasonal index for season i.

Econometric Forecasting Models

Distributed Lag Models

Y_t = a₀ + b₀ X _t +  λY_t-1  +  u_t

where λ is on the interval (0,1)  
 

e_t = r e_t-1  +  u_t    (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* = Y­_t – Y_t-1

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

2 autoregressive terms,  1 1^st 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(B^s) / Φ(B) Φ(B^s) Δ^dΔ^s  ] *   a_t 

where  

B = backshift operater such that BY = Yt-1

θ(B) = MA(q)

θ(B^s) = seasonal MA(Q)

Φ(B) = AR(p)

Φ(B^s) = seasonal AR(P)

Δ^d=  differencing

Δ^s = seasonal differencing

a_t = white noise term

References:

Using Econometrics: A Practical Guide (4th Edition)

A.H. Studenmund