代写Analyzing Time Series: Basic Regression Analysis

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    代写Analyzing Time Series: Basic Regression Analysis
    From Chapter 10 you will learn
    *The nature of time series data
    *Finite sample properties of OLS under classical assumptions
    *Functional forms and dummy variables
    *Trends and seasonality
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    *The nature of time series data
    *Temporal ordering of observations; may not be arbitrarily reordered
    *Typical features: serial correlation/nonindependence of observations
    *How should we think about the randomness in time series data?
    •The outcome of economic variables (e.g. GNP, Dow Jones) is uncertain; they should therefore be modeled as random variables
    •Time series are sequences of r.v. (= stochastic processes)
    •Randomness does not come from sampling from a population
    •“Sample“ = the one realized path of the time series out of the many possible paths the stochastic process could have taken
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    *Example: US inflation and unemployment rates 1948-2003
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    *Examples of time series regression models
    *Static models
    *In static time series models, the current value of one variable is modeled as the result of the current values of explanatory variables
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    *Examples for static models
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    *Finite distributed lag models
    *In finite distributed lag models, the explanatory variables are allowed to influence the dependent variable with a time lag
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    *Example for a finite distributed lag model
    *The fertility rate may depend on the tax value of a child, but for biological and behavioral reasons, the effect may have a lag
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    *Interpretation of the effects in finite distributed lag models
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    *Effect of a past shock on the current value of the dep. variable
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    *Graphical illustration of lagged effects
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    *Finite sample properties of OLS under classical assumptions
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    *Assumption TS.1 (Linear in parameters)
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    *Assumption TS.2 (No perfect collinearity)

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    *Notation
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    *Assumption TS.3 (Zero conditional mean)
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    *Discussion of assumption TS.3
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    *Strict exogeneity is stronger than contemporaneous exogeneity
    *TS.3 rules out feedback from the dep. variable on future values of the explanatory variables; this is often questionable esp. if explanatory variables „adjust“ to past changes in the dependent variable
    *If the error term is related to past values of the explanatory variables, one should include these values as contemporaneous regressors
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    *Theorem 10.1 (Unbiasedness of OLS)
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    *Assumption TS.4 (Homoscedasticity)
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    *A sufficient condition is that the volatility of the error is independent of the explanatory variables and that it is constant over time
    *In the time series context, homoscedasticity may also be easily violated, e.g. if the volatility of the dep. variable depends on regime changes
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    *Assumption TS.5 (No serial correlation)
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    *Discussion of assumption TS.5
    *Why was such an assumption not made in the cross-sectional case?
    *The assumption may easily be violated if, conditional on knowing the values of the indep. variables, omitted factors are correlated over time
    *The assumption may also serve as substitute for the random sampling assumption if sampling a cross-section is not done completely randomly
    *In this case, given the values of the explanatory variables, errors have to be uncorrelated across cross-sectional units (e.g. states)

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    *Theorem 10.2 (OLS sampling variances)
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    *Theorem 10.3 (Unbiased estimation of the error variance)
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    *Theorem 10.4 (Gauss-Markov Theorem)
    *Under assumptions TS.1 – TS.5, the OLS estimators have the minimal variance of all linear unbiased estimators of the regression coefficients
    *This holds conditional as well as unconditional on the regressors
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    *Assumption TS.6 (Normality)
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    *Theorem 10.5 (Normal sampling distributions)
    *Under assumptions TS.1 – TS.6, the OLS estimators have the usual nor-mal distribution (conditional on    ). The usual F- and t-tests are valid.
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    *Example: Static Phillips curve
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    *Discussion of CLM assumptions
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    *Discussion of CLM assumptions (cont.)
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    *Example: Effects of inflation and deficits on interest rates
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    代写Analyzing Time Series: Basic Regression Analysis

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    *Discussion of CLM assumptions
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    *Discussion of CLM assumptions (cont.)
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    *Using logarithmic functional forms 
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    *Interpretation
    *A higher (US) mimium wage lowers the employment rate.
    *US GNP does not seem to be important.
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    *Using dummy explanatory variables in time series
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    *Interpretation
    *During World War II, the fertility rate was temporarily lower
    *It has been permanently lower since the introduction of the pill in 1963
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    *Time series with trends
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    *Modelling a linear time trend
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    *Modelling an exponential time trend
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    *Example for a time series with an exponential trend
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    *Using trending variables in regression analysis
    *If trending variables are regressed on each other, a spurious re-  lationship may arise if the variables are driven by a common trend
    *In this case, it is important to include a trend in the regression
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    *Example: Housing investment and prices
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    *Example: Housing investment and prices (cont.)
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    *When should a trend be included?
    *If the dependent variable displays an obvious trending behaviour
    *If both the dependent and some independent variables have trends
    *If only some of the independent variables have trends; their effect on the dep. var. may only be visible after a trend has been substracted
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    *A Detrending interpretation of regressions with a time trend
    *It turns out that the OLS coefficients in a regression including a trend are the same as the coefficients in a regression without a trend but where all the variables have been detrended before the regression
    *This follows from the general interpretation of multiple regressions
    *Computing R-squared when the dependent variable is trending
    *Due to the trend, the variance of the dep. var. will be overstated
    *It is better to first detrend the dep. var. and then run the regression on all the indep. variables (plus a trend if they are trending as well)
    *The R-squared of this regression is a more adequate measure of fit
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    *Modelling seasonality in time series
    *A simple method is to include a set of seasonal dummies:
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    *Similar remarks apply as in the case of deterministic time trends
    *The regression coefficients on the explanatory variables can be seen as the result of first deseasonalizing the dep. and the explanat. variables
    *An R-squared that is based on first deseasonalizing the dep. var. may better reflect the explanatory power of the explanatory variables
    代写Analyzing Time Series: Basic Regression Analysis
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