Time Series Analysis

Trend Line - OLS

•  It has four major assumptions
• line
• normal distribution
• equal variance
• serial correlation (looking at the area of volatility)
• if not linear - transformation should occur. and anything transformed should revert it back to the original.

Autoregression

• regressing data against itself  - lags (for an example of how to in excel, see autoregression.xlsm)
• Sample model: $AR(2): y_t = b_0+b_1y_{t-1}+b_2y_{t-2}+ \text{error}$

Moving Average

• Sample model: $MA(2): y_t = e_0 +a_1e_{t-1}+a_2e_{t-2}$
• the $e$'s are the errors of an internal model.

ARMA

• Combination of both AR and MA Model
• Sample model: $ARMA(2,2): y_t =b_0+b_1y_{t-1}+b_2y_{t-2}+ e_0 +a_1e_{t-1}+a_2e_{t-2}$
• $AR(1):  y_t =b_0+b_1y_{t-1}+\text{error} $
• $b_1 <1$ - no problem. It stays stationary even if there are shocks.
• $b_1=1$ - shocks has permanent effect in the time series model.
• $b_1>1$ - not stationary. It explodes or decays.
• Covariance stationary
• mean is constant over time
• variance and covariance is constant over time
• $MA(1): y_t = e_0 +a_1e_{t-1}$
• the errors are artifacts of another model and its always brand new.
• Box-Jenkins
• AR
• ACF - decay to 0
• PACF -drops to 0
• MA
• ACF - drops to 0
• ACF - decay to 0
• Parsimony - important, how good is the forecast?
• AIC and BIC - choosing the model with the lowest value
• Random walk
• Model: $y_t = y_{t-1} + e$
• if the first differences is a white noise process.

Integrated Model

• If we difference it twice, we must go back twice after.
• ARIMA(p, d, q) model - AR(p), integration at d, MA(q)

Business Modelling

• MA - Kathang isip lamang because of the errors
• AR - is based on observable values
• Rule of thumb - 50 observations

Random walk with Drift

• An AR model:  $AR(2): y_t = b_0+b_1y_{t-1}+b_2y_{t-2}+ \text{error}$
• it is a random walk with trend

Mean Reversion

• MPT Application: Variance = $w^T \sigma w$ (sensitivity analysis on variancesensitivity.xlsx)
• must be homoscedastic
• not work on negatives

GARCH

• Conditional Heteroscedasticity applies
• Model:  $GARCH(1,1): h_t = \alpha_0+\alpha_1r_{t-1}^2+\beta_1h_{t-1}$

EWMA