INTRODUCTION ECONOMETRICS

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INTRODUCTION ECONOMETRICS

• Lecture week 13
• Time Series Econometrics Forecasting

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Forecasting

• Important part of econometrics analysis
• Various methods of forecasting

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Approaches to Economic Forecasting

• Exponential Smoothing Methods
• Single-Equation Regression Models
• Simultaneous- Equation Regression Models
• Autoregressive Integrated Moving Average (ARIMA)
  Models
• Vector Autoregressive Models (VAR)

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AR, MA, and ARIMA modeling of time series

• AR
• (Yt -?) a1(Yt-1 - ?) ut ..
  ..AR(1)
• (Yt -?) a1(Yt-1 - ?) a2(Yt-2 - ?) ut
  ...AR(2)
• (Yt -?) a1(Yt-1 - ?) a2(Yt-2 - ?)
  ak(Yt-k - ?)..AR(k)
• MA
• Yt ß0ut ß1ut-1.........
  MA(1)
• Yt ß0ut ß1ut-1 ß2ut-2......
  ..MA(2)
• Yt ß0ut ß1ut-1 ß2ut-2 ßkut-k
  ..........MA(k)
• ARMA
• Yt ? a1Yt-1 ß0ut ß1ut-1 ....A
  RIMA(1)
• Yt ? a1Yt-1 a2Yt-2 aKYt-K ß0ut ß1ut-1
  ß2ut-1 ßKut-K ....ARMA (p,q)

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The Box-Jenkins (BJ) Methodology

• Identification
• To find the appropriate values of p and q
• Correlogram and partial correlogram
• Estimation
• Parameters of the AR and MA
• Linear (OLS) models
• Nonlinear models
• Diagnostic checking
• Check if the residuals from this model are white
  noise
• Forecasting
• Superior compared to traditional econometric
  models

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Identification

• AR (p)
• ACF Decays exponentially or with damped sine
  wave pattern or both
• PACF Significant spikes through lags p
• MA (q)
• ACF Significant spikes through lags q
• PACF Declines exponentially
• ARMA (p, q)
• Exponential decay exponential decay

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Volatility of Financial Time Series

• Absolute Percent Change
• Variance of Spot Price around its trend
• Moving Average of Standard Deviation
• Autoregressive Conditional Heteroscedasticity
  (ARCH)

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Autoregressive Conditional Hetroscedasticity
(ARCH) Steps followed

• Most frequently used
• ARIMA(p,d,q) modeling
• Box-Jenkins (BJ) Methodology
• Test for ARCH effect
• ?2 a1 a2 ?2t-1 ?2t-2 ?2t-k
• Generalized Autoregressive Conditional
  Heteroscedasticy (GARCH) modeling.
• ?t2 a1 a2 ?2t-1 a3 ?2t-1

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Steps in the Box-Jenkins Methodology to model
ARIMA(p,q) process
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Autoregressive Conditional Hetroscedasticity
Effect (ARCH) (cont)
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ARCH/GARCH Approach to measure volatility in SAs
exchange rate Non stationarity confirmed)
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Sufficiency of First Differencing to convert
exchange rate to stationary series confirmed
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Step 1 Identification of p q in the ARMA
process- ARIMA(1,1,0) process identified
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Step 2 Estimation of ARMA(1,1,0) process
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Step 3 Diagnostic test using ARIMA(1,1,0) model.
Adequacy of the model confirmed
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Test for ARCH effect conducted the presence of
ARCH(1) confirmed i.e. conditional variance or
volatility of exchange rate is not constant.
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GARCH modeling conducted the significance of the
GARCH(1) variable confirmed overtime change in
the volatility of exchange rate
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Conditional Standard deviation (a measure of
volatility in the SAs exchange rate) frequency
of SAs exchange rate volatility has increased
since end of 2001 it is increasing.