Estimation and Inference of Linear Trend Slope Ratios with an Application to Global Temperature Data Tim Vogelsang & Nasreen Nawaz Department of Economics, Michigan State University Feb 8, 2017 Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 1
Differential Temperature Trends: Surface vs. Lower Troposphere Recent debate in empirical climate literature about temperature trends at surface and in lower troposphere. Climate models tend to have more warming in troposphere than at surface. Klotzbach et al (2009, JGR) provide empirical evidence that temperature trends in lower troposphere are lower than at surface especially over land. The ratio of troposphere temperature trends to surface temperature trends is labeled the " amplification ratio ". Estimation and inference of amplification ratios (trend ratios) is an interesting time series econometrics/statistics methodological topic. Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 2
Data Used by Klotzbach et al (2009) Updated to 2014 Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 3
Data Used by Klotzbach et al (2009) Updated to 2014 Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 4
Data Used by Klotzbach et al (2009) Updated to 2014 Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 5
Differential Temperature Trends: Surface vs. Lower Troposphere Klotzbach et al (2009) was controversial. Online debate in November, 2011 about how to best estimate a trend ratio. Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 6
Estimation of a Trend Ratio Simple time series model: y 1 t = µ 1 + β 1 t + u 1 t , (1) y 2 t = µ 2 + β 2 t + u 2 t , (2) for t = 1 , 2 , ..., T : The parameter of interest is the ratio of the two trend slopes: θ = β 1 β 2 � = 0 . , β 2 One side in the debate: θ should be estimated by regressing y 1 t on an intercept and y 2 t . Call this estimator � θ . Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 7
Estimation of a Trend Ratio Other side of the debate: use the ratio of OLS trend slopes: � β 1 � θ = � β 2 where � β 1 and � β 2 are the OLS estimators from (1) and (2). There was much online discussion about the two estimators with good intuition from some participants. What was missing is a rigorous analysis of the statistical properties of � θ and � θ . Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 8
Empirical Amplification Ratios Klotzbach et al Data (1979-2014) Land+Ocean Land Ocean � � � � � � Trend Ratio θ θ θ θ θ θ UAH/NCDC .934 .904 .583 .694 1.133 1.010 RSS/NCDC .912 .829 .623 .643 1.067 .952 UAH/HADC .883 .847 .708 .704 .941 .857 RSS/HADC .862 .777 .745 .652 .889 .808 NCDC/UAH .713 1.106 1.061 1.441 .545 0.990 NCDC/RSS .752 1.207 1.127 1.556 .552 1.050 HADC/UAH .762 1.180 .980 1.420 .626 1.166 HADC/RSS .804 1.288 1.024 1.534 .636 1.238 � θ is from regression of y 1 t on y 2 t ; � θ is ratio of OLS slope estimators. Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 9
Questions to Address How can � θ < 1 for both the regression of y 1 t on y 2 t and the regression of y 2 t on y 1 t ? Which estimator is better, � θ or � θ ? Do the magnitudes of β 1 and β 2 matter? How do we carry out inference about θ ? How should we construct confidence intervals for θ ? The online debate didn’t say much about inference. Those suggesting the use of � θ seemed to be using the usual OLS standard errors. OLS standard errors are not valid when data has serial correlation . Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 10
Slope Ratio Estimators in a Unified Framework Rewrite the regression equation for y 1 t as y 1 t = µ 1 + β 1 ( β 2 t ) + u 1 t = µ 1 + θ ( β 2 t ) + u 1 t . β 2 Rewrite the regression equation for y 2 t as β 2 t = y 2 t − µ 2 − u 2 t . Plug β 2 t into the equation for y 1 t to give y 1 t = µ 1 + θ ( y 2 t − µ 2 − u 2 t ) + u 1 t . Simple algebra gives y 1 t = δ + θ y 2 t + ǫ t ( θ ) , (3) where δ = µ 1 − θµ 2 and ǫ t ( θ ) = u 1 t − θ u 2 t . The regression of y 1 t on y 2 t does estimate θ . Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 11
A Problem with OLS The regression is y 1 t = δ + θ y 2 t + ǫ t ( θ ) , (3) where ǫ t ( θ ) = u 1 t − θ u 2 t and y 2 t = µ 2 + β 2 t + u 2 t . Obviously u 2 t is correlated with itself and u 1 t and u 2 t are correlated with each other in the temperature application. Problem: ǫ t ( θ ) is correlated with y 2 t in regression (3). This will cause � θ to be biased. If the magnitude of β 2 relative to the variation in u 2 t is large, this bias will be very small and won’t matter. If the magnitude of β 2 relative to the variation in u 2 t is small, this bias will be large and will matter. In the online discussion those that suggested � θ intuitively understood the bias problems with � θ . Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 12
Regressions with Errors Correlated with Regressors Regression models with errors correlated with the regressors have been studied in depth in the econometrics literature for 60+ years. Common solution: Instrumental Variables (IV) Estimation. Need a variable correlated with y 2 t but not correlated with ǫ t ( θ ) . Such a variable is called a "valid instrument" for y 2 t . t is the obvious candidate for an instrument for y 2 t . Note that if β 2 is small, then the correlation between t and y 2 t will be low - the "weak instrument" problem. Instrumental Variables with weak instruments leads to poor estimators of regression parameters. Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 13
Instrumental Variables Estimation Using t as instrument for y 2 t in regression (3) leads to the estimator T ∑ ( t − t )( y 1 t − y 1 ) � t = 1 θ = T ∑ ( t − t )( y 2 t − y 2 ) t = 1 T where t = T − 1 ∑ t is the sample average of time. t = 1 Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 14
Instrumental Variables Estimation Simple algebra gives � � − 1 T T ∑ ( t − t ) 2 ∑ ( t − t )( y 1 t − y 1 ) � β 1 t = 1 t = 1 � = � θ = = θ . � � − 1 � T T β 2 ∑ ∑ ( t − t ) 2 ( t − t )( y 2 t − y 2 ) t = 1 t = 1 The Instrumental Variables estimator of regression (3) is simply the ratio of OLS trend slopes. Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 15
Bias as a Function of Slope Magnitudes Using well known properties of OLS and IV in models where the regression error is correlated with regressors, we should expect the following to hold: Large β 2 : little bias in � θ and � θ with � θ and � θ giving similar numbers. Medium β 2 : noticeable bias in � θ , little bias in � θ . Small β 2 : substantial bias in � θ , some bias in � θ . Very small β 2 : both � θ and � θ essentially useless as estimators of θ . Magnitudes of β 2 are measured relative to the variation in u 2 t . Statistical theory in the paper shows that � θ , the ratio of estimated trend slopes, is the prefered estimator. Statistical theory shows how to compute a valid standard error for � θ . Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 16
Hypothesis Testing Could use � θ and its standard error to construct a t -statistic for testing hypotheses about θ . Confidence intervals for θ would be computed in the usual way. This standard approach won’t work well when trend slopes are small. Better approach is to write hypotheses about θ in terms of linear restrictions on β 1 and β 2 (Fieller’s method). Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 17
Hypothesis Testing Suppose we want to test the simple hypothesis H 0 : θ = θ 0 H 1 : θ = θ 1 � = θ 0 . We can write this hypothesis in terms of β 1 and β 2 as β 1 H 0 : = θ 0 , β 2 β 1 : = θ 1 � = θ 0 . H 1 β 2 or equivalently in terms of a linear restriction on β 1 and β 2 as H 0 : β 1 − β 2 θ 0 = 0 , H 1 : β 1 − β 2 θ 0 = β 2 ( θ 1 − θ 0 ) � = 0 . Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 18
Fieller’s Method Given a null value, θ 0 , construct the time series z t ( θ 0 ) = y 1 t − θ 0 y 2 t , where it follows from (1) and (2) that z t ( θ 0 ) = π 0 ( θ 0 ) + π 1 ( θ 0 ) t + ǫ t ( θ 0 ) , (4) with π 0 ( θ 0 ) = µ 1 − θ 0 µ 2 and π 1 ( θ 0 ) = β 1 − θ 0 β 2 . We can test the original null hypothesis by testing H 0 : π 1 ( θ 0 ) = 0 in (4) against the alternative H 1 : π 1 ( θ 0 ) � = 0 using π 1 ( θ 0 ) � � t θ 0 = � − 1 , (5) � 2 T � ∑ t = 1 ( t − t ) 2 λ θ 0 where � π 1 ( θ 0 ) is the OLS estimator from (4). Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 19
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