Introduction to time series

• What is the interpretation of coefficients in a simple linear regression (SLR) model?

  Coefficients in SLR represent the change in the dependent variable for a one-unit change in the independent variable. Example: If β₁ = 0.5, then for every one-unit increase in X, Y increases by 0.5.
• Define SST, SSR, and SSE in the context of regression analysis.


  SST (Total Sum of Squares) measures the total variation in the dependent variable.

  SSR (Regression Sum of Squares) measures the variation explained by the regression model.

  SSE (Error Sum of Squares) measures the unexplained variation or error in the model.
• How do you calculate the coefficient of determination (R²) using SST, SSR, and SSE?

  R^2=SST/SSR​

  It represents the proportion of the total variation in the dependent variable explained by the regression model.
• Explain the assumptions of simple linear regression (SLR) or multiple linear regression (MLR).

  Linearity: The relationship between the independent and dependent variables is linear.

  Independence: Observations are independent of each other.

  Homoscedasticity: Constant variance of errors across all levels of the independent variable(s).
• What distinguishes a biased estimator from an unbiased estimator?


  A biased estimator systematically overestimates or underestimates the parameter it estimates, while an unbiased estimator has an expected value equal to the true parameter.
• Define omitted variable bias and its implications in regression analysis.


  Omitted variable bias occurs when a relevant variable is not included in the regression model, leading to biased estimates of coefficients.
• Explain the F-test and its application in regression analysis.


  The F-test assesses the overall significance of the regression model by comparing the variance explained by the model to the unexplained variance. If the F-statistic is large and the associated p-value is small, the model is considered significant.
• What is a time series dataset and how does it differ from a cross-sectional dataset?

  A time series dataset consists of observations collected over time, while a cross-sectional dataset consists of observations collected at a single point in time.
• Define lags (Ytj) in the context of time series analysis.

  Lags represent past values of a variable Y at time t, denoted as Yt-j, where j is the number of periods back in time.
• Explain the concept of first-order differencing (∆Yt).

  First-order differencing (∆Yt) refers to the calculation of the difference between consecutive observations of a time series, often used to stabilize the variance or remove trends.
• What distinguishes a static time series model from a dynamic time series model?

  In a static time series model, the variables are not dependent on past values, while in a dynamic model, they are influenced by past observations.
• Describe the finite distributed lag model.

  The finite distributed lag model accounts for the effects of past values of the independent variable on the current value of the dependent variable, with a finite number of lagged terms.
• What are the assumptions TS1-TS3 in ordinary least squares (OLS) estimation for time series data?

  TS1: Linearity of the model, TS2: No perfect multicollinearity, TS3: Zero conditional mean (E[εt|Xt] = 0).
• Explain contemporaneous exogeneity in the context of time series analysis.


  Contemporaneous exogeneity means that the current values of the independent variables are not correlated with the error term in the regression model.
• What properties of the OLS estimators can be obtained under assumptions TS4-TS5?

  Under TS4: Homoscedasticity (constant variance of error terms) and TS5: No autocorrelation of error terms, OLS estimators are unbiased and consistent.
• How do you conduct hypothesis testing using t-tests in time series analysis?

  You use t-tests to determine whether the coefficient estimates of individual variables are significantly different from zero. The null hypothesis usually assumes that the coefficient is zero, and the alternative hypothesis assumes it's not.
• Define spurious correlation and explain how it can be detected using time series variables.

  Spurious correlation occurs when two variables appear to be correlated but actually have no causal relationship. It can be detected by examining correlations between variables and checking for common trends or patterns in time series plots.
• What is omitted variable bias according to slide 11?


  Omitted variable bias occurs when a relevant variable is not included in a regression model, leading to biased estimates of the coefficients of other variables.
• How can trends be observed in a time plot?

  Trends in a time plot can be observed by visually inspecting the data for systematic upward or downward movements over time.
• Explain the relationship between trends and omitted variable bias.

  Omitted variable bias can occur if a trend variable is omitted from the regression model, leading to biased estimates of other coefficients.
• When do we include a trend variable in our model?

  We include a trend variable in our model when there is evidence of a systematic and consistent change in the data over time that may affect the dependent variable.
• Define seasonality and provide examples.

  Seasonality refers to patterns or fluctuations in data that occur at regular intervals within a year. Examples include sales of ice cream increasing in summer and decreasing in winter, or holiday shopping peaks in December.
• How can we adjust for seasonality in regression analysis?

  We can adjust for seasonality by including seasonal dummy variables in the regression model, which capture the effects of different seasons on the dependent variable.
• Provide an example of spurious correlation in economics.


  Example: There may be a high correlation between ice cream sales and swimming pool accidents. However, this correlation is spurious as it is driven by a common factor, such as hot weather, rather than a direct causal relationship between the two variables.
• Explain the concept of omitted variable bias with an example.


  Example: In a regression analysis of wage determinants, omitting education level as a variable can lead to omitted variable bias if education level is correlated with both wages and other included variables like work experience.
• How can the presence of trends affect regression analysis?

  The presence of trends can lead to biased estimates of coefficients in regression analysis if trend variables are omitted, as the trend may explain part of the variation in the dependent variable.
• Give an example of a trend observed in economic data.


  Example: Long-term growth in GDP over several decades can be observed as a trend in economic data.
• Explain why it's important to include seasonal dummy variables in regression models.

  It's important to include seasonal dummy variables to account for the effects of seasonality on the dependent variable, ensuring that seasonal variations do not bias the estimates of other coefficients in the model.
• How can you visually detect seasonality in time series data?

  Seasonality can be visually detected by plotting the data over time and observing recurring patterns or fluctuations that occur at regular intervals within each year.
• Describe a situation where omitting a relevant variable could lead to spurious correlation.


  Example: Omitting inflation rate as a variable in a study of stock market returns and house prices could lead to spurious correlation, as both house prices and stock returns may be influenced by inflation.
• Explain the difference between a trend and seasonality in time series data.

  A trend represents a long-term systematic change or movement in the data over time, while seasonality refers to short-term fluctuations or patterns that recur at regular intervals within each year.
• Define stationarity in time series analysis. Give examples of strict and weak stationarity.

  Stationarity refers to a statistical property where the mean, variance, and covariance of a time series do not change over time. Strict stationarity implies that the entire distribution of the series remains constant, while weak stationarity (or covariance stationarity) requires only that the mean and covariance are constant.
• What is weak dependence in time series analysis?

  Weak dependence refers to the condition where the correlation between distant observations in a time series diminishes as the gap between them increases. It implies that the observations are not highly correlated, allowing for valid statistical inference.
• Explain the assumptions TS1'-TS3' in time series analysis.

  TS1' assumes that the series is weakly stationary, TS2' states that the series is weakly dependent, and TS3' implies that the errors are independent and identically distributed (i.i.d.). These assumptions are crucial for ensuring the validity of time series models.
• Define a white noise process and list its properties.

  A white noise process is a sequence of uncorrelated random variables with zero mean and constant variance. Its properties include independence of observations, zero autocovariance at all lags except at lag 0, and constant variance.
• What are the properties of the MA(1) process?

  In an MA(1) process, the current observation depends linearly on the current and one past error term. Its properties include zero mean, constant variance, and autocovariance that is zero for lags greater than 1.
• Explain the conditions for stationarity in the AR(1) process. Derive the mean and variance of the AR(1) process.

  For an AR(1) process to be stationary, the absolute value of the autoregressive coefficient must be less than 1. The mean of the AR(1) process is μ / (1 - φ), where μ is the mean of the error term and φ is the autoregressive coefficient. The variance is σ^2 / (1 - φ^2), where σ^2 is the variance of the error term.
• Define a unit root process and list its properties.


  A unit root process is a time series with a root of 1 in its autoregressive characteristic equation. Its properties include non-stationarity, meaning the series does not have a constant mean or variance, and a tendency for shocks to have permanent effects on the series.
• Explain the concept of unbiasedness and how strict exogeneity relates to it.

  Unbiasedness refers to the property of an estimator where, on average, it produces parameter estimates that are equal to the true values. Strict exogeneity ensures that the error term in a regression model is uncorrelated with all explanatory variables, which is necessary for unbiased estimation.
• Discuss the consequences of violating strict exogeneity on the unbiasedness of estimators.

  Violation of strict exogeneity can lead to biased estimators, meaning that the estimated coefficients will systematically overestimate or underestimate the true parameter values. This bias occurs because the error term is correlated with one or more explanatory variables.