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Forecasting and Time Series Analysis
source link: http://econometricsense.blogspot.com/2011/01/forecasting-and-time-series-analysis.html
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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
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}
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.
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
Distributed Lag Models
Yt = a0 + b0 X t + λYt-1 + ut
where λ is on the interval (0,1)
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:
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
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:
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 ] * at
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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