EC208 – Introductory Econometrics

Topic: Spurious/Nonsense Regressions (as part of chapter on Dynamic Models)

EC208 – Introductory Econometrics Topic: Spurious/Nonsense Regressions (as part of chapter on Dynamic Models) Important topic: - highlights pitfalls of regression analysis - data mining - stresses the importance of careful consideration of the statistical properties of economic time series

EC208 – Introductory Econometrics Topic: Spurious/Nonsense Regressions (as part of chapter on Dynamic Models) Important topic: - highlights pitfalls of regression analysis - data mining - stresses the importance of careful consideration of the statistical properties of economic time series Presentation: 1) Motivate the problem empirically 2) Provide simple technical arguments 3) Corroborate with simulation-based example 4) Give practical reccomendations

EC208 – Introductory Econometrics Spurious Regressions Telecommunications demand as a function of…

EC208 – Introductory Econometrics Spurious Regressions Telecommunications demand as a function of… the price of beef !!!???

EC208 – Introductory Econometrics Spurious Regressions Telecommunications demand as a function of… the price of beef !!!??? Argument: improved fit (R-squared)

EC208 – Introductory Econometrics Spurious Regressions Telecommunications demand as a function of… the price of beef !!!??? Argument: improved fit (R-squared) Another example: UK government expenditures and Private Consumption in Burkina-Faso (annual data, ), regression in logs

============================================================= Dependent Variable: Log(Gov) Method: Least Squares Sample: Included observations: 45 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C Log(Cons) ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criteri Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 3 EC208 – Introductory Econometrics Spurious Regressions

- Both series display a trending pattern, as most macroeconomic time series do

- What are the implications of this trending behaviour?

EC208 – Introductory Econometrics Spurious Regressions In a regression model is consistent if

EC208 – Introductory Econometrics Spurious Regressions In a regression model is consistent if must be well-behaved, so that “good behaviour” means that regressors must be stationary (and ergodic)

EC208 – Introductory Econometrics Spurious Regressions Strict stationarity implies that the distribution of a process does not change with time Weak stationarity: first and second moments are constant through time

EC208 – Introductory Econometrics Spurious Regressions Strict stationarity implies that the distribution of a process does not change with time Weak stationarity: first and second moments are constant through time Simple example: white noise

EC208 – Introductory Econometrics Spurious Regressions Strict stationarity implies that the distribution of a process does not change with time Weak stationarity: first and second moments are constant through time Simple example: white noise Many economic time series seem to be stationary after differencing:

EC208 – Introductory Econometrics Spurious Regressions Strict stationarity implies that the distribution of a process does not change with time Weak stationarity: first and second moments are constant through time Simple example: white noise Many economic time series seem to be stationary after differencing: This means that the series in levels is a random walk (or I(1) process):

EC208 – Introductory Econometrics Spurious Regressions 1 Random walk with drift and with no drift

EC208 – Introductory Econometrics Spurious Regressions Random walks arise from theory (e.g., Hall, 1978) or empirically A random walk is a special case of an autoregression Running an autoregression for each series…

============================================================= Dependent Variable: Log(Gov) Method: Least Squares Sample: Included observations: 45 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C LOG(GOV(-1)) ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 3 EC208 – Introductory Econometrics Spurious Regressions

============================================================= Dependent Variable: Log(Cons) Method: Least Squares Sample: Included observations: 45 ============================================================= Variable Coefficient Std. Error t-Statistic Prob. ============================================================= C LOG(Cons(-1)) ============================================================= R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) ============================================================= 3 EC208 – Introductory Econometrics Spurious Regressions

However, random walks are not stationary!

EC208 – Introductory Econometrics Spurious Regressions However, random walks are not stationary! - consider - substituting back, we get - substituting recursively, we get

EC208 – Introductory Econometrics Spurious Regressions However, random walks are not stationary! - consider - substituting back, we get - substituting recursively, we get The process has a deterministic trend component,, and a stochastic trend

EC208 – Introductory Econometrics Spurious Regressions However, random walks are not stationary! - consider - substituting back, we get - substituting recursively, we get The process has a deterministic trend component,, and a stochastic trend Shocks have permanent effects, long memory process

EC208 – Introductory Econometrics Spurious Regressions However, random walks are not stationary! - consider - substituting back, we get - substituting recursively, we get The process has a deterministic trend component,, and a stochastic trend Shocks have permanent effects, long memory process Taking expectations, we have

EC208 – Introductory Econometrics Spurious Regressions However, random walks are not stationary! - consider - substituting back, we get - substituting recursively, we get The process has a deterministic trend component,, and a stochastic trend Shocks have permanent effects, long memory process Taking expectations, we have Calculating the variance

EC208 – Introductory Econometrics Spurious Regressions Mean and variance vary with time! Random walks are non-stationary processes

EC208 – Introductory Econometrics Spurious Regressions Mean and variance vary with time! Random walks are non-stationary processes Back to OLS: non-stationary regressors means that does not converge to a finite matrix

EC208 – Introductory Econometrics Spurious Regressions Mean and variance vary with time! Random walks are non-stationary processes Back to OLS: non-stationary regressors means that does not converge to a finite matrix Hence, OLS is not consistent Nor asymptotically normal

EC208 – Introductory Econometrics Spurious Regressions Simple Monte Carlo experiment (Monte Carlo corresponds to a lab experiment in Econometrics…) - Generate n.i.d. variables - Construct independent random walks (no drift) - Regress - Compute - Repeat the process times

Tt-ratio rejection frequency (5%) Conventional t-tests reject the null of no relationship more often than it should (sample size worsens the problem) - R-squared converges to a non-degenerate distribution - DW statistic converges to 0 - This suggests informal way of recognizing spurious regressions - This experiment closely follows Granger & Newbold (1974), whose results were later confirmed analitically by Phillips (1986)

EC208 – Introductory Econometrics Spurious Regressions Practical implications Standard inference does not apply Importance of distinguishing between I(0) and I(1) What should we do to avoid nonsense regressions - regressions in differences: non-stationarity solved, but long run information (in levels) is lost -Cointegration: some economic variables share a common stochastic trend, implying a long run relationship (regression in levels makes sense, under certain conditions)

EC208 – Introductory Econometrics Spurious Regressions Students were introduced to new concepts: - Spurious/Nonsense regressions - stationary time series - Random walks/I(1) processes - Monte Carlo simulation - Hints for cointegration While making use of acquired ideas: -Regression/Large sample properties of OLS -Hypothesis testing -Fit of a regression -Autocorrelation (testing for…) -Dynamic modelling

EC208 – Introductory Econometrics Spurious Regressions The End