Estimation and Inference of Linear Trend Slope Ratios with an - - PowerPoint PPT Presentation

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: y1t = µ1 + β1t + u1t, (1) y2t = µ2 + β2t + u2t, (2) for t = 1, 2, ..., T: The parameter of interest is the ratio of the two trend slopes: θ = β1 β2 , β2 = 0. One side in the debate: θ should be estimated by regressing y1t on an intercept and y2t. 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:

• θ =
• β1
• β2

where β1 and β2 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 y1t on y2t;

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 y1t on y2t and the regression of y2t on y1t? Which estimator is better, θ or θ? Do the magnitudes of β1 and β2 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 y1t as y1t = µ1 + β1 β2 (β2t) + u1t = µ1 + θ (β2t) + u1t. Rewrite the regression equation for y2t as β2t = y2t − µ2 − u2t. Plug β2t into the equation for y1t to give y1t = µ1 + θ (y2t − µ2 − u2t) + u1t. Simple algebra gives y1t = δ + θy2t + ǫt (θ) , (3) where δ = µ1 − θµ2 and ǫt (θ) = u1t − θu2t. The regression of y1t on y2t 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 y1t = δ + θy2t + ǫt (θ) , (3) where ǫt (θ) = u1t − θu2t and y2t = µ2 + β2t + u2t. Obviously u2t is correlated with itself and u1t and u2t are correlated with each other in the temperature application. Problem: ǫt (θ) is correlated with y2t in regression (3). This will cause θ to be biased. If the magnitude of β2 relative to the variation in u2t is large, this bias will be very small and won’t matter. If the magnitude of β2 relative to the variation in u2t 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 y2t but not correlated with ǫt (θ). Such a variable is called a "valid instrument" for y2t. t is the obvious candidate for an instrument for y2t. Note that if β2 is small, then the correlation between t and y2t 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 y2t in regression (3) leads to the estimator

• θ =

T

∑

t=1

(t − t)(y1t − y1)

T

∑

t=1

(t − t)(y2t − y2) where t = T −1

T

∑

t=1

t is the sample average of time.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 14

Instrumental Variables Estimation

Simple algebra gives

• θ =
• T

∑

t=1

(t − t)2 −1

T

∑

t=1

(t − t)(y1t − y1)

• T

∑

t=1

(t − t)2 −1

T

∑

t=1

(t − t)(y2t − y2) =

• β1
• β2

= θ. 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 β2: little bias in θ and θ with θ and θ giving similar numbers. Medium β2: noticeable bias in θ, little bias in θ. Small β2: substantial bias in θ, some bias in θ. Very small β2: both θ and θ essentially useless as estimators of θ.

Magnitudes of β2 are measured relative to the variation in u2t. 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 β1 and β2 (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 H0 : θ = θ0 H1 : θ = θ1 = θ0. We can write this hypothesis in terms of β1 and β2 as H0 : β1 β2 = θ0, H1 : β1 β2 = θ1 = θ0.

• r equivalently in terms of a linear restriction on β1 and β2 as

H0 : β1 − β2θ0 = 0, H1 : β1 − β2θ0 = β2 (θ1 − θ0) = 0.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 18

Fieller’s Method

Given a null value, θ0, construct the time series zt (θ0) = y1t − θ0y2t, where it follows from (1) and (2) that zt (θ0) = π0(θ0) + π1(θ0)t + ǫt (θ0) , (4) with π0(θ0) = µ1 − θ0µ2 and π1(θ0) = β1 − θ0β2. We can test the original null hypothesis by testing H0 : π1(θ0) = 0 in (4) against the alternative H1 : π1(θ0) = 0 using tθ0 =

• π1(θ0)
• λ

2 θ0

• ∑

T t=1(t − t)2

−1 , (5) where π1(θ0) is the OLS estimator from (4).

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 19

Fieller’s Method

Because of serial correlation in the data, we use λ

2 θ0 in place of the

usual OLS error variance estimator.

• λ

2 θ0 is a robust variance estimator computed as

• λ

2 θ0 =

γθ0

0 + 2 T −1

∑

j=1

k j M

• γθ0

j ,

• γθ0

j

= T −1

T

∑

t=j+1

• ǫt (θ0)

ǫt−j (θ0) ,

• ǫt (θ0) = zt (θ0) −

π0(θ0) − π1 (θ0) t. k(x) is a weighting function and M is a bandwidth parameter. Larger values of M place relatively higher weight on high lags of γθ0

j .

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 20

Fieller’s Method

It is not difficult to show that

• π1(θ0) =

β1 − θ0 β2. Therefore, tθ0 is testing H0 : β1 β2 = θ0 using β1 and β2 in a linear manner rather than using the ratio, θ. Avoiding the ratio is what makes tθ0 a well behaved test statistic even for small or zero slope values. Notice that testing H0 : θ = 1 with tθ0 is computationally equivalent to testing H0 : β1 − β2 = 0 using β1 and β2. tθ0 automatically handles serial correlation in u1t and u2t and allows them to be correlated.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 21

Confidence Intervals

A 100(1 − α)% confidence interval for θ is the collection of values for θ0 that cannot be rejected at the α significance level using the tθ0 statistic, i.e. the values of θ0 such |tθ0| ≤ cvα/2. A confidence interval is really a non-rejection region for θ. We need to find the values of θ0 such that

• π1(θ0)
• λ

2 θ0

• ∑

T t=1(t − t)2

−1

• ≤ cvα/2.

The calculation is tricky because λ

2 θ0 depends on the value of θ0.

In standard settings the variance estimator is exactly invariant to the parameter of interest leading to the usual confidence interval formula.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 22

Confidence Intervals

We show in paper that the confidence interval for θ can take one of three possible forms:

1

r1 ≤ θ0 ≤ r2

2

θ0 ≤ r1 and θ0 ≥ r2

3

−∞ < θ0 < ∞

r1 and r2 are the ordered real roots of a second order polynomial (r1 ≤ r2). The first confidence interval occurs when trend slopes are large. The second confidence interval occurs when trend slopes are medium to small. The third confidence interval occurs when trend slopes are very small

• r zero.

Regardless of the case, the confidence interval always contains θ because π1( θ) = β1 − θ β2 = 0 and tθ0 = 0 for θ0 = θ.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 23

Practical Performance

The paper has a Monte Carlo simulation study to assess practical performance of the various estimators and t-statistics. Asymptotic theory is a good guide to what happens in practice.

• θ is the least biased estimator and it has good precision.

The tθ0 approach to testing works very well for any slope magnitudes. Traditional t-statistics based on θ and θ do not work well when trend slopes are medium or small.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 24

Practical Recommendations

We recommend that θ be used to estimate θ and that confidence intervals be reported using tθ0. Switching y1t with y2t in regression (3) has no real effects for θ. The slope ratio estimate is inverted as are the end points of the confidence interval. Switching y1t with y2t in regression (3) can have substantial effects

• n

θ because of the correlation between y1t and u2t − β2

β1 u1t and the

associated bias in θ. The recommended approach is essentially the approach used by Klotzbach et al (2009) with two minor differences:

They report results for specific hypotheses about θ; no confidence intervals. They use a variance estimator that is only robust to red noise serial correlation.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 25

Back to Empirical Application

Klotzbach et al Data (1979-2014) Land+Ocean Land Ocean Series

• β
• β
• β

UAH .133 [.075, .192] .180 [.099, .260] .107 [.054, .159] RSS .122 [.063, .182] .166 [.080, .252] .101 [.052, .149] NCDC .148 [.107, .188] .259 [.190, .327] .106 [.074, .138] HADC .158 [.115, .200] .255 [.185, .326] .125 [.092, .157] Trend Ratio

• θ

θ0 = 1.2

• θ

θ0 = 1.1

• θ

θ0 = 1.6 UAH/NCDC .904 [.644, 1.108] .694 [.479, .865] 1.01 [.680, 1.251] RSS/NCDC .829 [.564, 1.003] .643 [.401, .813] .952 [.683, 1.124] UAH/HADC .847 [.602, 1.040] .704 [.486, .883] .857 [.555, 1.078] RSS/HADC .777 [.522, 0.950] .652 [.410, .823] .808 [.550, 0.981] Note: 95% confidence intervals in brackets using Daniell k(x) function with b = 0.1.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 26

Back to Empirical Application

Klotzbach et al Data (1979-2008) Land+Ocean Land Ocean Series

• β
• β
• β

UAH .134 [.053, .216] .178 [.072, .284] .109 [.032, .186] RSS .142 [.063, .221] .205 [.099, .310] .111 [.042, .180] NCDC .164 [.113, .215] .289 [.205, .374] .117 [.074, .160] HADC .176 [.123, .229] .290 [.205, .375] .134 [.090, .179] Trend Ratio

• θ

θ0 = 1.2

• θ

θ0 = 1.1

• θ

θ0 = 1.6 UAH/NCDC .818 [.454, 1.028] .615 [.347, .772] .930 [.415, 1.208] RSS/NCDC .862 [.536, 1.055] .707 [.452, .887] .948 [.558, 1.149] UAH/HADC .763 [.418, 0.966] .614 [.343, .775] .810 [.342, 1.079] RSS/HADC .805 [.488, 1.003] .705 [.447, .891] .826 [.457, 1.033] Note: 95% confidence intervals in brackets using Daniell k(x) function with b = 0.1.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 27

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 28

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 29

Simulations

Data generating process: y1t = β1t + u1t, y2t = β2t + u2t, u1t = 0.4u2t + 0.3u1t−1 + ε1t, u2t = 0.5u2t−1 + ε2t, [ε1t, ε2t] ∼ i.i.d. N(0, I2), u10 = u20 = 0. µ1 = 0, µ2 = 0 without loss of generality β1 = θβ2 T = 50, 100, 200 10, 000 replications 5% nominal level for tests

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 30

Some Formal Theoretical Results

Work out asymptotic properties of θ and θ under the assumption that u1t and u2t are covariance stationary and weakly dependent. Assume that T −1/2

[rT ]

∑

t=1

u1t u2t

• ⇒ ΛW (r) ≡

B1(r) B2(r)

• ,

(6) where r ∈ [0, 1], [rT] is the integer part of rT and W (r) is a 2 × 1 vector of independent standard Wiener processes. Λ is not necessarily diagonal allowing for correlation between u1t and u2t both contemporaneously and across time.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 31

Some Formal Theoretical Results

Given the definition of ǫt (θ), it immediately follows from (6) that T −1/2

[rT ]

∑

t=1

ǫt (θ) ⇒ λθw(r), (7) where w(r) is a univariate standard Wiener process and λ2

θ =

• 1

−θ

• ΛΛ

1 −θ . λ2

θ is the long run variance of ǫt (θ):

λ2

θ = γ0 + 2 ∞

∑

j=1

γj, γj = E (ǫt(θ)ǫt−j(θ)) .

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 32

Asymptotic Results

The following hold as T → ∞ where in all cases θ = β1/β2. Large Trend Slopes (β1 = β1, β2 = β2): T 3/2

• θ − θ
• d

→ N

• 0, 12λ2

θ/β 2 2

• ,

T 3/2

• θ − θ
• d

→ N

• 0, 12λ2

θ/β 2 2

• .

Medium Trend Slopes (β1 = β1T −1/2, β2 = β2T −1/2): T

• θ − θ
• d

→ N

• 12E [u2tǫt(θ)] /β

2 2, 12λ2 θ/β 2 2

• ,

T

• θ − θ
• d

→ N

• 0, 12λ2

θ/β 2 2

• .

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 33

Asymptotic Results

Small Trend Slopes (β1 = β1T −1, β2 = β2T −1):

• θ − θ
• p

→ 1 12β

2 2 + E

• u2

2t

−1 E [u2tǫt(θ)] , √ T

• θ − θ
• d

→ N

• 0, 12λ2

θ/β 2 2

• .

Very Small Trend Slopes (β1 = β1T −3/2, β2 = β2T −3/2):

• θ − θ
• p

→

• E
• u2

2t

−1 E [u2tǫt(θ)] ,

• θ − θ
• ⇒

1 12β2 +

1

0 (s − 1

2)dB2(s) −1 λθ

1

0 (s − 1

2)dw(s).

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 34

Summary of Asymptotic Results

Summary of large T results:

• θ and

θ are equally good when trends slopes are large.

• θ and

θ are poor estimators (inconsistent) when trends slopes are very small.

• θ remains a good estimator when trend slopes are medium sized

whereas θ becomes biased and inconsistent.

Across the board, θ is the preferred estimator of θ. The paper proposes a bias corrected version of θ which improves performance of θ when trend slope are medium. Unfortunately, this bias correction does not work well for small or very small trend slopes. We do not know of an estimator for linear trend ratios that is better than θ.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 35

Fieller’s Method: Asymptotics

Suppose that M = bT where b ∈ (0, 1]. Then as T → ∞, the following holds for any values of β1 and β2: tθ0 ⇒ Z

• Pb(Q(r))

, Z ∼ N(0, 1), where Q(r) = w(r) − rw(1) − 6r(r − 1) 1

• s − 1

2

• dw(s) and Pb(·)

depends on the functional form of k(x). For example, for the Bartlett kernel we have k(x) = 1 − x, |x| ≤ 0 0, |x| > 1, Pb(Q(r)) = 2 b 1

0 Q (r)2 dr −

1−b

Q (r + b) Q (r) dr

• .

Critical values for tθ0 are nonstandard and have been simulated by Bunzel and Vogelsang (2005, Journal of Business and Economic Statistics).

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 36

Asymptotics for t-statistics

Theorem

(Large Trend Slopes) Suppose that (7) holds. Let M = bT where b ∈ (0, 1] is fixed. Let β1 = β1, β2 = β2 where β1, β2 are fixed with respect to T, and let θ1 = θ0 + T −3/2θ∆. Then as T → ∞, tOLS, tBC , tIV ⇒ Z

• Pb(Q(r)) +

β2θ∆

• 12λ2

θ1Pb(Q(r))

, tθ0 ⇒ Z

• Pb(Q(r)) +

β2θ∆

• 12λ2

θ0Pb(Q(r))

, where Z ∼ N(0, 1), Q(r) = w(r) − 12L(r) 1

• s − 1

2

• dw(s),
• w(r) = w(r) − rw(1), L(r) = r
• s − 1

2

• ds and Z and Q(r) are

independent.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 37

Asymptotics for t-statistics

Theorem

(Medium Trend Slopes): Suppose that (7) holds. Let M = bT where b ∈ (0, 1] is fixed. Let β1 = T −1/2β1, β2 = T −1/2β2 where β1, β2 are fixed with respect to T, and let θ1 = θ0 + T −1θ∆. Then as T → ∞, tOLS ⇒ Z

• Pb(H1(r)) + 12β

−1 2 E(u2tǫt (θ))

• 12λ2

θ1Pb(H1(r))

+ β2θ∆

• 12λ2

θ1Pb(H1(r))

, tBC , tIV ⇒ Z

• Pb(Q(r)) +

β2θ∆

• 12λ2

θ1Pb(Q(r))

, tθ0 ⇒ Z

• Pb(Q(r)) +

β2θ∆

• 12λ2

θ0Pb(Q(r))

, where H1(r) = Q(r) − 12

• λθ1β2

−1 L(r) · E(u2tǫt (θ)).

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 38

Asymptotics for t-statistics

Theorem

(Small Trend Slopes): Suppose that (7) holds. Let M = bT where b ∈ (0, 1] is fixed. Let β1 = T −1β1, β2 = T −1β2 where β1, β2 are fixed with respect to T, and let θ1 = θ0 + T −1/2θ∆. Then as T → ∞, tOLS, tBC

d

→ 1

• β

2 2Pb (L(r))

• β

2 2

1

0 (s − 1 2)2ds + E (u2 2t)

−1 , tIV ⇒ Z

• Pb(Q(r)) +

β2θ∆

• 12λ2

θ1Pb(Q(r))

, tθ0 ⇒ Z

• Pb(Q(r)) +

β2θ∆

• 12λ2

θ0Pb(Q(r))

.

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 39

Asymptotics for t-statistics

Theorem

(Very Small Trend Slopes): Suppose that (6) and (7) hold. Let M = bT where b ∈ (0, 1] is fixed. Let β1 = T −3/2β1, β2 = T −3/2β2 where β1, β2 are fixed with respect to T, and let θ1 = θ0 + θ∆. Then as T → ∞, T −1/2tOLS, T −1/2tBC =

• E(u2

2t)

−1 E(u2tǫt(θ)) + θ∆

• Pb (H2(r)) [E (u2

2t)]−1

, tIV ⇒ 1

0 (s − 1 2)dw(s) +

• β2

1

0 (s − 1 2)2ds + 1 0 (s − 1 2)dB2(s)

• λ−1

θ θ∆

• Pb (H3(r))

1

0 (s − 1 2)2ds

, tθ0 ⇒ Z

• Pb(Q(r)) +

β2θ∆

• 12λ2

θ0Pb(Q(r))

,

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 40

Asymptotics for t-statistics

Where H2(r) = w(r) −

• β2

1

0 (s − 1

2)2ds +

1

0 (s − 1

2)dB2(s) −1 ×

1

0 (s − 1

2)dw(s)

• β2L(r) +

B2(r)

• ,

H3(r) = w(r) − 12

• β2L(r) +

B2(r)

• ×
• β2 + 12

1

• s − 1

2

• dB2(s)

−1 1

• s − 1

2

• dw(s),
• B2(r) = B2(r) − rB2(1)

Tim Vogelsang (MSU) Ratios of Trend Slopes (Santa Fe Climate Conference) Feb 8, 2017 41