Carbon and Temperature

STA 542 · Homework 6 workbook

Atmospheric carbon dioxide and global temperature have both risen. Does past carbon concentration help predict temperature once we account for their trends? How does the answer depend on what we remove?

The data file contains annual observations for 1959–2023. Use co2_ppm, the Mauna Loa atmospheric CO₂ concentration in parts per million, and choose one temperature record: temperature_global_c (land and ocean), temperature_land_c, or temperature_ocean_c. All three are global averages, measured in °C relative to the 1901–2000 average. Use the same choice throughout.

(a) Remove the trend

Compare four ways to transform each recorded series \(v_t\): \[ \begin{aligned} \text{Linear detrending:}\quad &v_t^{L}=v_t-(\widehat\alpha_1+\widehat\beta_1 t),\\ \text{Quadratic detrending:}\quad &v_t^{Q}=v_t-(\widehat\alpha_2+\widehat\beta_2 t+\widehat\gamma_2t^2),\\ \text{First differences:}\quad &\Delta v_t=v_t-v_{t-1},\\ \text{Second differences:}\quad &\Delta^2v_t=v_t-2v_{t-1}+v_{t-2}. \end{aligned} \] Inspect the supplied plots: which methods leave a visible trend? Choose two methods worth comparing in the rest of the exercise, and explain what a transformed temperature observation measures under each. Apply each chosen method to both coordinates. For a chosen detrending method, fit its polynomial separately to each series by least squares against year, using all 65 observations.

CO2 and temperature in levels, after linear detrending, and after quadratic detrending. Axes give calendar years and ppm or degrees Celsius.

Annual CO₂ and global land-and-ocean temperature, 1959–2023. Top: observations with fitted linear and quadratic trends. Middle: linear residuals. Bottom: quadratic residuals. Real data: NOAA GML and NOAA NCEI.

CO2 and temperature in levels, first differences, and second differences. Axes give calendar years and ppm or degrees Celsius.

Annual CO₂ and global land-and-ocean temperature. Top: observations. Middle: first differences. Bottom: second differences. Differences are dated by the ending year, starting in 1960 and 1961 respectively. Real data: NOAA GML and NOAA NCEI.

CO2 and temperature in levels, after linear detrending, and after quadratic detrending. Axes give calendar years and ppm or degrees Celsius.

Annual CO₂ and global land temperature, 1959–2023. Top: observations with fitted linear and quadratic trends. Middle: linear residuals. Bottom: quadratic residuals. Real data: NOAA GML and NOAA NCEI.

CO2 and temperature in levels, first differences, and second differences. Axes give calendar years and ppm or degrees Celsius.

Annual CO₂ and global land temperature. Top: observations. Middle: first differences. Bottom: second differences. Differences are dated by the ending year, starting in 1960 and 1961 respectively. Real data: NOAA GML and NOAA NCEI.

CO2 and temperature in levels, after linear detrending, and after quadratic detrending. Axes give calendar years and ppm or degrees Celsius.

Annual CO₂ and global ocean temperature, 1959–2023. Top: observations with fitted linear and quadratic trends. Middle: linear residuals. Bottom: quadratic residuals. Real data: NOAA GML and NOAA NCEI.

CO2 and temperature in levels, first differences, and second differences. Axes give calendar years and ppm or degrees Celsius.

Annual CO₂ and global ocean temperature. Top: observations. Middle: first differences. Bottom: second differences. Differences are dated by the ending year, starting in 1960 and 1961 respectively. Real data: NOAA GML and NOAA NCEI.

(b) Fit the pair

For each transformed pair, consider the working model \[ \begin{pmatrix}Y_{C,t+1}\\Y_{T,t+1}\end{pmatrix} = \begin{pmatrix}a_C\\a_T\end{pmatrix} + \underbrace{\begin{pmatrix}A_{CC}&A_{CT}\\A_{TC}&A_{TT}\end{pmatrix}}_{A} \begin{pmatrix}Y_{C,t}\\Y_{T,t}\end{pmatrix} +Z_{t+1}, \] where \(Y_{C,t}\) and \(Y_{T,t}\) are the transformed CO₂ and temperature variables, and \(a=(a_C,a_T)^\top\). Assume a stable causal VAR(1) with iid \(N_2(0,\Sigma_Z)\) innovations and positive-definite \(\Sigma_Z\). We will assess this model in part (d).

Fit each row by least squares, including an intercept and both coordinates from the preceding year. Do this for your two chosen transformations and report \(\widehat a\) and \(\widehat A\) for each. There are 64 paired transitions after either detrending, 63 after first differences, and 62 after second differences.

R hint: Use resid(lm(...)) for detrending, I(year^2) for the quadratic term, and diff(..., differences = 2) for second differences. Follow the two-regression construction in Workbook 11, Example 11.8, A VAR(1) for log pelt records. Here, use the recorded units without logs.

(c) Read the predictive directions

Under the model in part (b), which entry of \(A\) must be zero for CO₂ not to Granger-cause temperature? Which entry gives the reverse condition? Granger causality here concerns the transformed coordinates.

For each fit, interpret \(\widehat A_{TC}\): holding the current transformed temperature fixed, how does a 1 ppm increase in the current transformed CO₂ change the fitted next-year temperature prediction? State the prediction target in each case.

The tables supply estimates and individual nominal 95% confidence intervals from ordinary rowwise regression (confint()). These are model-based approximations for the dynamic regressions and treat estimated trends as fixed. The 95% level applies to each entry separately.

Entry Estimate Lower Upper
\(A_{CC}\) 0.9813 0.9317 1.0310
\(A_{CT}\) 0.4177 −1.6051 2.4406
\(A_{TC}\) 0.0099 0.0033 0.0164
\(A_{TT}\) 0.1445 −0.1222 0.4113
Entry Estimate Lower Upper
\(A_{CC}\) 0.7441 0.5855 0.9026
\(A_{CT}\) 1.3612 0.0708 2.6516
\(A_{TC}\) −0.0004 −0.0349 0.0342
\(A_{TT}\) 0.1795 −0.1018 0.4609
Entry Estimate Lower Upper
\(A_{CC}\) 0.5143 0.2934 0.7352
\(A_{CT}\) 1.0459 −0.2923 2.3841
\(A_{TC}\) −0.0262 −0.0701 0.0178
\(A_{TT}\) −0.2581 −0.5241 0.0079
Entry Estimate Lower Upper
\(A_{CC}\) −0.5677 −0.7868 −0.3485
\(A_{CT}\) 1.8397 1.0731 2.6063
\(A_{TC}\) −0.1191 −0.1814 −0.0568
\(A_{TT}\) −0.3559 −0.5738 −0.1380
Entry Estimate Lower Upper
\(A_{CC}\) 0.9951 0.9440 1.0463
\(A_{CT}\) −0.3115 −1.4103 0.7874
\(A_{TC}\) 0.0254 0.0131 0.0376
\(A_{TT}\) −0.0803 −0.3437 0.1831
Entry Estimate Lower Upper
\(A_{CC}\) 0.8224 0.6664 0.9785
\(A_{CT}\) −0.0884 −0.7866 0.6099
\(A_{TC}\) 0.0032 −0.0563 0.0628
\(A_{TT}\) −0.0568 −0.3233 0.2098
Entry Estimate Lower Upper
\(A_{CC}\) 0.5770 0.3495 0.8045
\(A_{CT}\) −0.0831 −0.7437 0.5775
\(A_{TC}\) −0.0028 −0.0873 0.0817
\(A_{TT}\) −0.4690 −0.7143 −0.2236
Entry Estimate Lower Upper
\(A_{CC}\) −0.5045 −0.7596 −0.2493
\(A_{CT}\) 0.4655 0.0613 0.8697
\(A_{TC}\) −0.1096 −0.2467 0.0275
\(A_{TT}\) −0.5618 −0.7791 −0.3446
Entry Estimate Lower Upper
\(A_{CC}\) 0.9762 0.9318 1.0206
\(A_{CT}\) 1.5316 −0.7746 3.8378
\(A_{TC}\) 0.0040 −0.0008 0.0089
\(A_{TT}\) 0.3515 0.0991 0.6040
Entry Estimate Lower Upper
\(A_{CC}\) 0.6867 0.5438 0.8297
\(A_{CT}\) 2.9414 1.5773 4.3054
\(A_{TC}\) −0.0041 −0.0323 0.0242
\(A_{TT}\) 0.3840 0.1148 0.6532
Entry Estimate Lower Upper
\(A_{CC}\) 0.5099 0.3125 0.7072
\(A_{CT}\) 2.8339 1.2500 4.4178
\(A_{TC}\) −0.0298 −0.0628 0.0032
\(A_{TT}\) −0.0826 −0.3474 0.1822
Entry Estimate Lower Upper
\(A_{CC}\) −0.4540 −0.6363 −0.2716
\(A_{CT}\) 3.0717 2.1657 3.9778
\(A_{TC}\) −0.1019 −0.1449 −0.0590
\(A_{TT}\) −0.2658 −0.4793 −0.0524

For each of your two fits, which cross-coefficient intervals contain zero? Compare the transformations and state what these intervals support about Granger causality under the working model.

(d) Assess the model

The diagnostics below are supplied for all four methods. Each residual entry gives the sample autocorrelations at lags 1 and 2, in that order.

Transformation \(\rho(\widehat A)\) CO₂ residuals Temperature residuals
Linear detrending 0.986 0.550, 0.437 0.027, −0.202
Quadratic detrending 0.743 0.055, −0.111 0.026, −0.178
First differences 0.477 −0.111, −0.029 −0.072, −0.290
Second differences 0.649 −0.259, −0.268 −0.297, −0.273
Transformation \(\rho(\widehat A)\) CO₂ residuals Temperature residuals
Linear detrending 0.988 0.562, 0.441 0.000, −0.139
Quadratic detrending 0.822 0.073, −0.167 −0.008, −0.134
First differences 0.577 −0.132, −0.116 −0.161, −0.371
Second differences 0.578 −0.211, −0.435 −0.304, −0.395
Transformation \(\rho(\widehat A)\) CO₂ residuals Temperature residuals
Linear detrending 0.986 0.572, 0.469 0.090, −0.314
Quadratic detrending 0.640 0.083, −0.080 0.093, −0.291
First differences 0.271 −0.090, 0.014 0.011, −0.337
Second differences 0.659 −0.231, −0.307 −0.260, −0.275

Use the table for your chosen temperature record. In at most four sentences, compare your two fits and assess which you would prefer for a stationary VAR(1), if either. Use the plots and residual diagnostics to assess the fits and the credibility of their confidence intervals. Would this analysis estimate the physical effect of increasing atmospheric CO₂ on global temperature? Explain why or why not.

Write your answers by hand and append the two fitted models and the short R code used to obtain them. List collaborators and material assistance.

Data: NOAA GML and NOAA NCEI, downloaded 20 October 2026; see the accompanying data notes. The Mauna Loa record includes measurements from Maunakea during the December 2022–July 2023 interruption.