__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

where λ is on the interval (0,1)

_{t}= a_{0}+ b_{0}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

2 autoregressive terms, 1 1

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}= b_{0}+ b_{1}Y*_{t-1 }+ b_{2}Y*_{t-2 }+ b_{3}e_{t-1 }+ e_{t}2 autoregressive terms, 1 1

^{st}difference, 1 moving average termSeasonal 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^{d}Δ^{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}= differencingΔ

^{s}= seasonal differencinga

_{t }= white noise term

__References:__Using Econometrics: A Practical Guide (4th Edition)

A.H. Studenmund

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