Chapter 9: Time Series Analysis and Serial Correlation

Time Series Overview

  • Time Series Data: Involves a single entity over multiple points in time.
  • Notation Difference:
    • Cross-sectional: $Yi = b0 + b1 X{1i} + b2 X{2i} + b3 X{3i} + e_i$ where $i$ goes from 1 to N
    • Time series: $Yt = b0 + b1 X{1t} + b2 X{2t} + b3 X{3t} + e_t$ where $t$ goes from 1 to T

Characteristics of Time Series Analysis

  1. Fixed Order of Observations: Cannot reorder data.
  2. Smaller Samples: Time series samples are often smaller than cross-sectional.
  3. Complexity: Theory behind time-series analysis tends to be more complicated.
  4. Serial Correlation: The error term can be influenced by past events, known as serial correlation.

Pure Serial Correlation

  • Defined as a violation of Classical Assumption IV in a correctly specified equation.
  • First-order Serial Correlation:
    • Form: $et = ho e{t-1} + u_t$
    • Where:
    • $
      ho$: first-order autocorrelation coefficient
    • $u_t$: classical error term
  • Impact of $ ho$:
    • If $
      ho = 0$, no serial correlation.
    • As $
      ho$ approaches ±1, the degree of serial correlation increases.
    • Positive serial correlation ($
      ho > 0$) implies current errors are similar to past errors.
    • Negative serial correlation ($
      ho < 0$) implies current errors are opposite in sign to past errors.

Visualization of Serial Correlation

  • Positive Serial Correlation: Current observation has the same sign as the previous - often due to prolonged effects from external shocks.
  • No Serial Correlation: Error terms are uncorrelated with each other.
  • Negative Serial Correlation: Current observation tends to differ in sign from the previous.

Additional Forms of Serial Correlation

  • More complex forms exist beyond first-order, including:
    • Seasonal effects (e.g., $et = r e{t-4} + u_t$)
    • Involvement of multiple previous observations (e.g., $et = r1 e{t-1} + r2 e{t-2} + ut$)

Impure Serial Correlation

  • Caused by specification error that includes parts of the effect of misspecification in the error term.
  • Types of specification errors:
    1. Omitted Variable: Missed variables that affect the dependent variable.
    2. Incorrect Functional Form: Using a wrong functional form can create systematic biases in the residuals.

Consequences of Serial Correlation

  1. No bias in coefficient estimates from pure serial correlation.
  2. OLS is no longer the best linear estimator (minimum variance).
  3. OLS estimates can be biased, making hypothesis testing unreliable.

The Durbin-Watson Test

  • Purpose: To test for first-order serial correlation.
  • Assumptions:
    1. Includes an intercept in the model.
    2. Assumes first-order serial correlation.
    3. Does not include lagged dependent variables.
  • Statistic Formula:
    d=∑(e<em>t−e</em>t−1)2∑et2d = \frac{\sum (e<em>t - e</em>{t-1})^2}{\sum e_t^2}
  • Interpretation:
    • $d o 0$: Extreme positive serial correlation.
    • $d o 4$: Extreme negative serial correlation.
    • $d o 2$: No serial correlation.

Decision Rule for Durbin-Watson Test

  • For positive serial correlation:
    • If $d < dL$, reject $H0$.
    • If $d > dU$, do not reject $H0$.
    • $dL < d < dU$: inconclusive.

Lagrange Multiplier (LM) Test

  • Tests for serial correlation via lagged residuals.
  • Steps:
    1. Obtain residuals from the original equation.
    2. Specify auxiliary equation: $et = Yt - \hat{Y}_t$.
    3. Test whether coefficients of lagged residuals are significant.
  • Null hypothesis is rejected if significant.

Remedies for Serial Correlation

  1. Start with checking for possible specification errors.
  2. If pure serial correlation remains, consider:
    • Generalized Least Squares (GLS): Adjusts for pure serial correlation by transforming the model.
    • Newey-West Standard Errors: Adjust standard errors for serial correlation while keeping coefficients unchanged.

Generalized Least Squares (GLS)

  • Eliminates pure first-order serial correlation.
  • The transformation encompasses:
    Y<em>t∗=Y</em>t−rY<em>t−1,X</em>1∗=X<em>1t−rX</em>1t−1Y<em>t^* = Y</em>t - rY<em>{t-1}, \quad X</em>1^* = X<em>{1t} - rX</em>{1t-1}

Newey-West Standard Errors

  • Adjusts the estimated standard errors to account for serial correlation, typically yielding larger errors than OLS, thus lowering t-scores.
  • Essential for reliable model inference in the presence of serial correlation without biasing coefficients.