ARIMA ANALYSIS WITH INTERVENTIONS / OUTLIERS

Transcription

1 TASK Run intervention analysis on the price of stock M: model a function of the price as ARIMA with outliers and interventions. SOLUTION The document below is an abridged version of the solution provided to the client. The SAS output associated with the whole study is huge. Here we display only selected output for illustrative purposes. The objective is to give an idea of the types of analysis that this project required. ARIMA ANALYSIS WITH INTERVENTIONS / OUTLIERS Analysis Our study is done in several consecutive steps. Please see the SAS output, which is attached in blocks. If you need to learn the intervention analysis methodology, check out William W.S. Wei, Time Series Analysis, Univariate and Multivariate Methods. STEP 1: We analyze ACF / PACF / IACF of Price, daily differences of price (Diff) and daily log-returns (LogReturn). We also run augmented Dickey-Fuller tests for the three time series. The conclusions are: Price is clearly non-stationary, while Diff and LogReturn seem to be stationary. We choose Diff for subsequent ARMA modeling, as that may lead to a relatively simple and nice ARIMA model for Price. STEP 2: We experiment with ARMA models for Diff until the residuals exhibit the properties of white noise. Also, we use Akaike information criterion to identify the best structure (it is displayed in tables Minimum Information Criterion ). An MA(2) model for Diff seems to be the best fit. STEP 3: We identify 5 additive outliers (AO type in the language of the book). We add them to the ARMA model as additive shifts. The meaning of them can be additive interventions, related to stricter regulations etc. In particular, the shift that happened on April 9, 2008 is especially big. It may be related to FDA introducing new rules requiring additional tests for diabetes drugs (which are produce of company M). The estimates of the magnitudes of the additive shifts are contained in the output. STEP 4: Without the shifts, Diff may be described by an ARMA model. However, the structure of the model may be slightly different from that identified in step 2. Now outliers / shifts are not obscuring the true correlation picture. So we perform model identification again, making sure the new residuals exhibit the properties of white noise. The new optimal model for Diff turns out to be seasonal ARMA((11, 14, 18), 1) + AO-type shifts with pulse functions (see the output). Therefore the optimal model for Price is ARIMA((11, 14, 18), 1, 1) + AO-type shifts with step functions STEP 5: We perform forecasts for Diff based on this model.

2 Selected SAS Output STEP 1

3 STEP 2 ARIMA Estimation Optimization Summary Estimation Method Conditional Least Squares Parameters Estimated 3 Termination Criteria Maximum Relative Change in Estimates Iteration Stopping Value Criteria Value 2.86E-15 Maximum Absolute Value of Gradient R-Square Change from Last Iteration Objective Function Sum of Squared Residuals Objective Function Value Marquardt's Lambda Coefficient 1E12 Numerical Derivative Perturbation Delta Iterations 3 Warning Message Estimates may not have converged. Conditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > t Lag MU MA1, MA1, Constant Estimate Variance Estimate Std Error Estimate AIC SBC Number of Residuals 1163 * AIC and SBC do not include log determinant.

4 Correlations of Parameter Estimates Parameter MU MA1,1 MA1,2 MU MA1, MA1, Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations Model for variable Diff Estimated Mean Moving Average Factors Factor 1: B**(1) B**(2) STEP 3 Outlier Detection Summary Maximum number searched 5 Number found 5 Significance used 0.05 Outlier Details Approx Chi- Prob> Obs Time ID Type Estimate Square ChiSq APR-2008 Additive < SEP-2004 Additive < SEP-2004 Additive < AUG-2004 Additive < NOV-2006 Additive <.0001

5 STEP 4 Conditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > t Lag Variable Shift MU Price 0 MA1, Price 0 MA1, Price 0 NUM < outl933 0 NUM < outl45 0 NUM < outl40 0 NUM < outl8 0 NUM < outl575 0 Constant Estimate Variance Estimate Std Error Estimate AIC SBC Number of Residuals 1163 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Variable Price Price Price outl933 Parameter MU MA1,1 MA1,2 NUM1 Price MU Price MA1, Price MA1, outl933 NUM outl45 NUM outl40 NUM outl8 NUM outl575 NUM Correlations of Parameter Estimates Variable outl45 outl40 outl8 outl575 Parameter NUM2 NUM3 NUM4 NUM5 Price MU Price MA1, Price MA1, outl933 NUM outl45 NUM outl40 NUM outl8 NUM outl575 NUM Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

6 Model for variable Price Estimated Intercept Period(s) of Differencing 1 Moving Average Factors Factor 1: B**(1) B**(2) Input Number 1 outl933 Overall Regression Factor Input Number 2 outl45 Overall Regression Factor Input Number 3 outl40 Overall Regression Factor Input Number 4 outl8 Overall Regression Factor Input Number 5 outl575 Overall Regression Factor

7 Name of Variable = CleanDiff Mean of Working Series Standard Deviation Number of Observations 1163 Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq Autocorrelations Conditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > t Lag MU MA1, AR1, AR1, AR1, Constant Estimate Variance Estimate Std Error Estimate AIC SBC Number of Residuals 1163 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU MA1,1 AR1,1 AR1,2 AR1,3 MU MA1, AR1, AR1, AR1, Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

8 Autoregressive Factors Factor 1: B**(11) B**(14) B**(18) Moving Average Factors Factor 1: B**(1) STEP 5 Forecasts for variable CleanDiff Obs Forecast Std Error 95% Confidence Limits Statistical & Financial Consulting by Stanford PhD consulting@stanfordphd.com