Time Series_3_Generalisation

1. Linear and Log-Linear Trend Model

1.1 Assumptions: (1) E[ε] = 0;

　　　　　　　　  (2) ε is normally distributed;

　　　　　　　　  (3) E[ε²] = σ² const; and violation of this is called 'heteroskedasticity' 异方差, when subsamples spread out more than rest of the sample (conditional/unconditional);

 　　　　　　　　 (4) ε is iid; and violation is called 'autocorrelation' or 'serial correlation' (+/-);

　　　　　　　  　(5) independent variables is uncorrelated with residuals;

　　　　　　　  　(6) no linear relation between any two or more independent variables; and violation is called 'multicollinearity';

1.2 Deal with Heteroskedasticity

1.2.1 Scatter plots 

1.2.2 Breusch-Pagan chi-square test:

　　(1) BP statistic = n × R² (n = # of observations, df = k, k = # of independent variables, R²  is from a 2nd regression of ε² from the 1st regression on the independent variables) 

　　(2) This is regression of ε² on the independent variables, if conditional hete-ity exisits, the ind-variables will significantly contribute to explanation of ε².

　　(3) One-tailed test. Hete-ity is only problem if R² and BP test statistic are too large.

1.2.3 Correction

　　Calculate 'robust standard errors' (or 'white-corrected standard errors' or 'hete-ity consistent standard error') and use it recalculate the t-statistics using original regression coeff.

1.3 Deal with Autocorrelation

1.3.1 Residual plots

1.3.2 Durbin-Watson statistic

　　(1) DW ≈ 2(1 - r), where r is correlation coeff between residuals from one period and those from previous period;

　　(2) DW = 2, then ε terms are homoskedastic and not serially correlated, i.e. r = 0; 

　　　  DW > 2, means r < 0, thus ε terms nega-seri-correlated;

　　(3) DB Decision Rule: H₀ is 'There's no positive autocorrelation.'

　　　　　　　　　　　　Reject H₀ if DW < d_lower, and conclude positive autocorrelation;

　　　　　　　　　　　　Inconclusive between d_lower and d_upper;

　　　　　　　　　　　　Don't reject if DW >  d_upper;

1.3.3 Correction

　　(1) Adjust coeff standard errors using Hansen method;

　　(2) Improve specification of model;

1.4 Deal with Multicollinearity

1.4.1 Signal: When t-tests indicate 'none coeff significantly ≠ 0' while at same time F-test significant and R² is high.

1.4.2 Correction: Omit one or more independent variables using stepwise regression.

2. Forecasting ModelsAutoregressive (AR) Models

2.1 First Differencing: transform data to covariance stationary series when it's random walk (i.e. hs unit root).

2.1.1 Stationarity(history is relevant for forecasting):

　　　　　　　　Distribution doesn't change over time;

　　　　　　　　E(Y_t) and Var(Y_t) is const;

　　　　　　　　Cov(Y_t, Y_t-j) doesn't depend on t;

FirstAR model is correctly specified when no autocorrelation in ε terms, otherwise this AR need correction. 

2.2 Autoregressive Model AR(p)

2.1.1 Method one: Run AR model and examine autoc-ion.

2.1.2 Method two: Dickey Fuller test

2.1.3 Mean Reversion: if x_t > b₀ / (1 - b₁), the AR(1) predicts x_t+1 will be lower than x_t.

2.3 Autoregressive Distributed Lag Model ADL(p, r)

 

 

2.4 Model Selection

Criterion BIC (Bayesian Information Criterion) or AIC to decide optimal length of lag(p)

 

 

 Choose p that minimizes BIC.

3. Other Notations

3.1 Use Clustered HAC SE in panel data to correct for serial correlation, allowing errors to be correlated within a cluster but not across the clusters. While in time series data, use Newey-West HAC SE which estimates correlations between lagged values in time series.

Criterion:m (# of lags) = 0.75 T^1/3, T is # of obsevation (or periods)

3.2 Seasonality

3.3 ARCH

ARCH exists if E[ε²] in one period is dependent on E[ε²] of previous period. Thus, standard errors of regression coeff in AR models and hypothesis tests of these coeff are in valid.

3.4 Cointegration: Two time series are economically linked (related to the same macro variables) or follow the same trend (relationship is not expected to change).

The ε term from regression of these two time series will be covariance stationary and t-tests will be reliable.

y_t = b₀ + b₁x_t + ε

The ε term are tested for unit root using Dickey Fuller test with critical t-values calculated by Engle and Granger (DF-EG test). If rejects NULL of unit root, then ε terms generated by these two time series are covariance stationary and they are cointegrated, thus we can use the regression to model their relationship.