CO2 has significant implications for hourly ambient temperature: Evidence from Hawaii

A small group of climate scientists and influencers have vigorously disputed the scientific consensus on climate change. They have contributed to a belief system that has impeded policy actions to reduce emissions. They accept that more CO2 in the atmosphere has consequences for the climate but strongly deny that the magnitude of the effect is significant. Using hourly CO2 data from the Mauna Loa Observatory in Hawaii, this article examines whether the hourly temperature data at the nearby Hilo International Airport support this belief. ARCH/ARMAX methods are employed because the hourly temperature data, even in Hawaii, are both highly autoregressive and volatile. The temperature data are analyzed using an archive of day‐ahead hourly weather forecast data to control for expected meteorological outcomes. The model is estimated using 42,928 hourly observations from August 7, 2009, through December 31, 2014. CO2 concentrations are found to have statistically significant implications for hourly temperature. The model is evaluated using hourly data from January 1, 2015, through December 31, 2017. The findings add to the consilience of evidence supporting the scientific consensus on climate change.

points since March 2015. The survey also indicates that the percentage of Americans who believe that global warming is due mostly to natural environmental changes is 30%, which is only two percentage points lower than in 2015. While 65% of the respondents indicated they are worried about climate change, only 46% indicated that they have personally experienced its effects. At the sub-national level, the estimates range from 29% (Wyoming) to 61% (Washington, DC). In Alaska, a state where Thoman et al. (2020) have indicated that the effects of climate change are quite clear, the estimate of perceived personal climate impact is 41% . One possible contributing factor to this apparent lack of climate awareness is an unwillingness by conservatives to attribute extreme weather events to climate change (Carman et al., 2022). This finding is consistent with a 2019 survey indicating that 59% of registered voters in the United States believe climate change is mostly human-induced, only 25% of conservative Republicans endorse this view (Leiserowitz et al., 2019). Despite their minority status, the views of climate change deniers in the United States represent a formidable obstacle to adopting strict policies by the United States government to reduce greenhouse gases. For example, the climate-related legislation signed by President Biden on August 16, 2022 (Tankersley, 2022), was only approved by the narrowest of margins (in the U.S. Senate, the vote was 51 to 50; under the "normal" rules of the Senate, the legislation would have needed 60 votes to avoid a filibuster). The legislation represents a highly significant climate policy advance even though it largely leaves the emissions of CO 2 from power plants fueled by coal unregulated, a feature that the electric power industry is almost sure to exploit. Future legislation could resolve this and other shortcomings. However, their passage is highly uncertain given the stringent filibuster rules in the United States Senate and the determined opposition by Senators who reject the scientific consensus on climate change. Given that the United States is a leading emitter of these gases, this state of affairs reduces the effectiveness of international agreements to reduce emissions. Consistent with this view, Wynes (2022, p. 1404) have concluded that "the available evidence does not yet indicate that the world has seriously committed to achieving the 1.5 • C goal." In contrast to the multivariate statistical analyses such as Lean and Rind (2008), Rohde et al. (2012), Stern and Kaufmann (2014), and Ribes et al. (2017) that focus on the annual measures of global temperature, this article examines the effect of CO 2 on the hourly ambient temperature at a specific location. The central presumption of the approach is that the public is much more interested in weather than climate. The null hypothesis in this analysis is that CO 2 does not affect temperature. Regarding methodology, the analysis proceeds by including the weather conditions expected by meteorologists and other relevant factors as covariates. The implicit assumption associated with this approach is that meteorologists do not explicitly take CO 2 levels into account when forecasting weather. If they did, the issue of CO 2 's effect on hourly temperature would not be a matter of contention within the meteorological profession (Stenhouse et al., 2017).
It is acknowledged that some meteorologists and climate scientists, regardless of their views on climate change, are likely to be highly skeptical of the paper's findings. In terms of background, the reference works by von Storch and Zwiers (1999), Wilks (2019), and Delsole and Tippett (2022) largely limit their discussion of the autoregressive process to two lags without explanation, while a substantially longer process is considered in this analysis. The ARMAX (autoregressive-moving-average with exogenous inputs modeling) method, one of the core components of this paper's modeling approach, is not discussed by Wilks (2019) and Delsole and Tippett (2022). It is incompletely described by Von Storch and Zwiers (1999, p. 215). In all three reference works, there is no mention of the ARCH (autoregressive conditional heteroskedasticity) method, another important component of the approach presented here, even though this procedure is recognized in other sectors as being invaluable in modeling time series data that exhibit periods of high volatility followed by relative calm. This characteristic is clearly evident in the case of hourly temperature data, even in Hawaii. Given this background, atmospheric scientists without an appreciation of the autoregressive and heteroscedastic nature of hourly temperature and the methods to adequately analyze data with these characteristics may not be well-positioned to critique the analysis presented in this article. Those who assert that the methods employed in this article are not appropriate for addressing atmospheric environmental issues at the hourly level are cheerfully invited to inspect Table A1, which reports out-of-sample hour-ahead temperature predictions for Central Park in New York City (and other locations). These predictions for New York City are highly accurate, even though the only inputs are the lagged hourly temperatures at the Mauna Loa Observatory in Hawaii and the lagged hourly temperatures at Central Park in New York City. A discussion of how time-series methods make this possible is discussed in the Appendix.
The article is organized as follows. Section 2 presents data indicating that Hawaii's climate is changing and discusses climate deniers' views. A general approach to analyzing climate issues is also discussed. Section 3 discusses the data employed in this study. Section 4 presents more details on the modeling approach. Section 5 presents the ARCH/ARMAX model that includes CO 2 concentrations as an input. Section 6 discusses the estimation methods and presents the modeling results. Section 7 presents the results from an alternative formulation that does not consider CO 2 as an input. This model is found to be inferior to the model that includes CO 2 concentrations as a covariate. Section 8 presents an out-of-sample model evaluation. The evaluation indicates that the temperature predictions are accurate when the CO 2 levels from the current era are used as inputs in the estimated prediction equation but are inaccurate when the prevailing levels of CO 2 from the preindustrial era are assumed. Section 9 summarizes the findings and discusses a possible demonstration project that might help induce a new era regarding public attitudes and policies concerning climate change.

THE REALITY OF CLIMATE CHANGE AND THE BELIEFS OF CLIMATE CHANGE DENIERS
Numerous researchers have documented the reality of climate change. The most accessible studies include Wuebbles et al. (2017) andIPCC (2021). The literature review by Mudelsee (2019) concludes that the suspected global warming hiatus is not supported by statistical analysis of the data and that the upward trend in the global temperature is a real phenomenon.
In terms of magnitude, global temperature has risen by about 1.2 • C over the 1880-1900 average (Craigmile & Guttorp, 2022). Consistent with this finding, there is a significant upward trend in the reported temperature at the Manau Loa Observatory (MLO) in Hawaii from 1977 through 2020 ( Figure 1).
Some scientists accept that more CO 2 in the atmosphere has consequences for the climate but strongly deny that the magnitude of the effect is significant. In this article, these individuals will be identified as "Climate Deniers." Two members of this group are Richard Lindzen and William Happer. Both of these individuals have made substantial contributions to their respective fields. However, their views are an outlier in the field of climate change. In their words, "No scientist familiar with radiation transfer denies that more carbon dioxide is likely to cause some surface warming. But the warming would be small and benign." (Lindzen & Happer, 2021) Judith Curry also accepts the core principle of climate science but believes that the natural variability in the climate calls the scientific consensus into question. In her written testimony to the Committee on Science, Space, and Technology of the United States House of Representatives, she notes, "Recent data and research supports the importance of natural climate variability and calls into question the conclusion that humans are the dominant cause of recent climate change." Curry (2015) Unfortunately, Professor Curry's 2015 written statement to the United States House of Representatives does not provide citations to support her claim. A more recent non-peer-reviewed article on extreme weather events authored by Curry F I G U R E 1 The annual temperature at the Mauna Loa Observatory in Hawaii, 1977Hawaii, -2020. Source: Global Monitoring Division of the National Oceanic and Atmospheric Administration (NOAA). https://gml.noaa.gov/dv/data/index.php?site=MLO&parameter_name= Meteorology makes it clear that she has a very low opinion of a linkage between greenhouse gas concentrations and heat waves. In her words, "Heat waves are the new polar bears, stoking alarm about climate change. Climate scientists addressing this in the media are using misleading and/or inadequate approaches." (Curry, 2021).
John Christy accepts that more CO 2 in the atmosphere has consequences for the climate but asserts that the effect is small. Christy also asserts that the increase in greenhouse gas concentrations is not changing the weather system. In his words, "The weather we really care about isn't changing, and Mother Nature has many ways on her own to cause her climate to experience considerable variations in cycles. If you think about how many degrees of freedom are in the climate system, what a chaotic nonlinear, dynamical system can do with all those degrees of freedom, you will always have record highs, record lows, tremendous storms and so on. That's the way that system is." (Christy, 2019) Christy seems to be indicating that it is overly speculative to claim that greenhouse gases affect the weather and climate. A more recent blog by Christy goes further than this by concluding that anomalous weather events are largely unrelated to greenhouse gases. For example, he asserts that the 2021 heatwave in the Pacific Northwest was simply a "black swan" event (Christy, 2021).
In contrast to Christy's beliefs and the other views reported above, the more recently published research has indicated that virtually all of the global warming since 1950 can be attributed to human influences Ribes et al., 2017;Stocker et al., 2013). One possible objection to these findings is that they are largely based on simulated rather than real data. This objection may be understandable, but it is puzzling that the climate deniers discussed above can be so confident in their views given the unrefuted statistical findings by Lean and Rind (2008) that indicated that natural factors, including solar activity, are unable to explain the historical trend in annual temperature over the period 1889 to 2006. The statistical findings by Rohde et al. (2012), Stern and Kaufmann (2014), and Ribes et al. (2017) are also relevant. None of these findings have significantly altered the Climate Deniers' unsubstantiated and unwavering claim that the quantitative effect of CO 2 is small. While some climate scientists may believe that further improvements in the global climate models will resolve matters, this view could be considered naïve because the potential to create doubt will remain regardless of the magnitude of the modeling improvements. Drawing on Popper (2014), the fundamental problem that climate scientists face is that it is impossible to definitely "prove" that the scientific consensus is correct. As Lawson (2014) points out Climate scientists … "can put forward the evidence, but they cannot force their audience to agree with them. They can point to the fact that carbon dioxide is a greenhouse gas, that its levels in the atmosphere have risen by 40% since the Industrial Revolution, and that we can only account for the recent rise in global temperatures by including the enhanced greenhouse effect alongside known natural factors such as solar variability and ocean currents. They can point to the observed patterns of warming as consistent with warming due to greenhouse gases in contrast to other possible causes of warming. But in the end, the reasoning is inductive, not deductive. It is not proof ." (Lawson, 2014) Fortunately, while climate science cannot provide definitive proof, the contrary hypothesis, in the words of Popper (2014, p. 40-41), is "falsifiable," that is, it is possible to test the contrary hypothesis that CO 2 does not significantly affect temperature. In this article, this contrary claim is tested using hourly data, the bedrock of almost all the reported meteorological data, by accounting for nonlinearities and using methods that recognize the heretofore unrecognized but easily understood systematic nature of hourly temperature.

DATA
The study employs data on the hourly atmospheric concentration level of CO 2 as reported by Mauna Loa Observatory (MLO), an atmospheric station of the Earth System Research Laboratory (ESRL), Global Monitoring Division (GMD) of the National Oceanic and Atmospheric Administration (NOAA). Continuous atmospheric measurements of CO 2 have been recorded at this observatory since 1974 (Thoning et al., 2021). Tans and Thoning (2020) provide an overview of the methods used to collect and process the data and indicate that the CO 2 measurements made at MLO accurately reflect the  Figure 2). This phenomenon has been noted by Keeling et al. (1976), Thoning et al. (1989), and Bakwin et al. (1998). The variable of interest in this article is the hourly temperature at Hilo International Airport measured in degrees Celsius. The airport is located approximately 60 km from MLO. The data were obtained from NOAA's National Climatic Data Center (https://www.ncdc.noaa.gov/).
The study also uses day-ahead forecasted weather data for each hour of the day. Consistent with the dependent variable, the specific location corresponding to the weather forecasts is Hilo International Airport. The meteorological variables include hourly forecasted temperature, dewpoint, humidity, visibility, wind speeds, probability of precipitation, and measures of forecasted sky conditions. The data were obtained from CustomWeather, a California-based weather forecasting firm that generates forecasts for approximately 80,000 locations in 200 countries (http://customweather.com/). CustomWeather's forecasts are based on National Weather Service data and its own algorithms. The hourly forecasts employed in this study are generated at hour 19 UT (hour 18 UT when daylight savings time at CustomWeather's headquarters in California is in effect) the day before real-time. Specifically, the archived forecasts for day one of the sample for each hour were generated at hour 18 (or 19) on day zero of the sample. This approach is followed for each day of the sample.
Consideration was given to using the modeled temperature by ERA5 produced by the Copernicus Climate Change Service at ECMWF as the control measure. This option was not pursued once it was discovered that the ERA5 reanalysis temperature data corresponding to Hilo International Airport were less accurate than CustomWeather's day-ahead forecast over the same period. Consideration was also given to using the hourly day-ahead forecast data reported by the National Weather Service as controls. However, these were also less accurate than CustomWeather's day-ahead hourly forecasts.
Observations were dropped from the sample if the hourly carbon concentration data were missing, if data on the actual temperature were missing, or if the day-ahead forecast data were unavailable. In total, 4448 observations were deleted, with most deletions resulting from missing CO 2 data. The deletions left a final sample size of 42,928 observations.

AN APPROACH TO MODELING THE RELATIONSHIP BETWEEN CO 2 CONCENTRATIONS AND HOURLY AMBIENT TEMPERATURE
In addition to their findings on seasonality, Keeling et al. (1976), Thoning et al. (1989), and Bakwin et al. (1998) have established that there is significant diurnal variation in the hourly CO 2 concentration levels. Consistent with these findings, the autocorrelations have a distinct pattern ( Figure 3). Specifically, there is a pronounced "blip" in the autocorrelations

F I G U R E 4
The autocorrelations in hourly temperature at Hilo International Airport, August 7, 2009-December 31, 2014 at hour t − 25, t − 50, t − 75, t − 100 and so forth. One possible explanation for this phenomenon is the quality control check that is performed every 25 h. During this process, a target gas is run through the measurement system to ensure that the assigned values of the calibration mixtures are correct (Tans & Thoning, 2020, p. 3). While the "blip" in the autocorrelations is not evident if one considers the partial autocorrelations, the approach adopted here exploits the "blip" in the full hourly CO 2 autocorrelations by using the CO 2 concentration level in hour t − 25 as an explanatory variable in the analysis. This variable has a 0.9587 full correlation with the CO 2 level in hour t but is obviously itself unaffected by the temperature in hour t, given that it is lagged by 25 h (the partial autocorrelation is not employed in the analysis because, as discussed by Box and Jenkins (1976, pp. 64-65) it understates the full autocorrelation when the lag exceeds one because the reported partial correlation for lag k if k is greater than one does not reflect the correlations corresponding to previous lags). Thus, this lagged variable can provide evidence of a relationship between CO 2 and temperature not tainted by possible two-way causality between the two variables.
Concerning the temperature data, a key attribute that has implications for the modeling approach is the robustness of the autocorrelations in the hourly values ( Figure 4). As the figure indicates, the magnitude of the reported autocorrelative process over the first 24 hourly lags ranges from −0.3696 to 0.9324 (note: the value can be negative depending on the value of the covariance in temperature, one of the inputs in the calculated autocorrelation). This systematic pattern is so distinct that it calls into question the claims that the weather system in terms of hourly temperature is chaotic. It may also be reasonable to wonder if the data are nonstationary. This concept is briefly discussed by DelSole and Tippett (2022, p. 102), who assert that "most climate time series are nonstationary" but do not provide specifics. Specifically, readers are not informed that statistical analysis of nonstationary variables can yield spurious results (Kennedy, 2008, p. 301). The challenges posed by nonstationary variables are extensively discussed by Mudelsee (2014, pp. 246-284;Mudelsee, 2020, p. 38-45).
Following Kennedy (2008, p. 302), if the variable y t equals α y t-1 + ε t where ε t is a pure random error term, then the data series is nonstationary if α equals one. The data series is deemed stationary if the parameter α is less than one; the tests of nonstationary are thus known as unit root tests. The most well-known unit root test is the Augmented Dickey-Fuller test, in which the null hypothesis presumes nonstationary, that is, it presumes a unit root. In this case, the Augmented Dickey-Fuller test for a unit root in the hourly temperature data series yields a p-value less than 0.0001, indicating a very high level of support for rejecting the null hypothesis of a unit root. The p-value is less than 0.0001 when the Augmented Dickey-Fuller analysis considers either a drift or trend term. It is also less than 0.0001 when the Augmented Dickey-Fuller analysis excludes the constant term. The Phillips-Perron test for a unit root also strongly supports rejecting the null hypothesis of a unit root. Both of these tests also strongly support rejecting the null hypothesis of a unit root when applied to all of the covariates (e.g., hourly CO 2 levels). These findings are not limited to hourly temperature data at Hilo International Airport. Unit root tests were performed on all the temperature data sets employed in the Appendix. In every case, the null hypothesis of a unit root in hourly temperature was rejected using the Augmented Dickey-Fuller and Phillips-Perron methodologies. The application of the DF-GLS methodology developed by Elliott et al. (1996) was considered based on its power, as reported by Baum and Hurn (2021, p. 120). The test's null hypothesis is that the variable of interest has a random walk, possibly with drift. The test's default alternative hypothesis is that the variable of interest is stationary about a linear time trend. This test requires no gaps in the data, a condition that is not satisfied for any of the variables considered in this study. Applying this test to the ERA5 hourly reanalysis temperature data corresponding to Hilo International Airport, over January 1, 1985, through December 31, 2019, a series without gaps, yields p values that unambiguously reject the null hypothesis of a unit root. Applied to the gap-free data series of annual temperature at MLO from 1977 to 2020, the test rejects the null hypothesis of a unit root at the five percent level. In short, the ERA5 hourly data and the MLO annual temperature data are trend stationary, meaning that differencing, the standard solution to the unit root issue, is not necessarily required to achieve stationary (Kennedy, 2008, p. 308). Statistical analysis not tainted by unit-root issues is possible if one includes a trend term, CO 2 concentrations being an obvious candidate. In contrast, the DF-GLS methodology applied to the annual global temperature anomalies reported by the HadCRUT data set (https:// www.metoffice.gov.uk/hadobs/hadcrut5/data/current/download.html) unambiguously fails to reject the null hypothesis of a unit root which poses a challenge to meaningful statistical analysis.
While the unit root results for the hourly and annual local temperature data are encouraging, statistical analysis of the local annual temperature data is highly problematic because of degrees of freedom issues. Focusing instead on hourly temperature would seem to be the most appropriate research path but it is not without its challenges. Specifically, a quantitative analysis of hourly time-series temperature data needs to control for its autocorrelative nature to effectively extract the CO 2 "signal" from the "noise" in the data. The method of ordinary least squares presumes an absence of autocorrelation; thus, it is highly deficient. This point of caution is consistent with Granger and Newbold (1974, p. 117), who conclude the following: "In our opinion the econometrician can no longer ignore the time series properties of the variables with which he [or she] is concerned-except at his [or her] peril." The consequences of ignoring their assessment includes suboptimal forecasts and invalid tests of statistical significance. To avoid these deficiencies, the estimation process employed in this article presumes that the temperature during hour t is not independent of the temperature outcomes in previous hours. The analysis also considers the heteroskedastic nature of hourly temperature, a condition in which the volatility in temperature is not constrained to be constant.

A CO 2 -AUGMENTED ARCH/ARMAX MODEL OF TEMPERATURE
As suggested in the previous section, the modeling approach employed in this article presumes that lagged temperatures influence the current hourly temperature. The approach also enables modeling the volatility in temperature. The approach is entitled "A CO 2 -augmented autoregressive conditional heteroskedasticity/ autoregressive-moving-average with exogenous inputs model of temperature" (hereafter, a CO 2 -augmented ARCH/ARMAX model of temperature). The ARCH specification is employed to model conditional heteroskedasticity, an outcome in which the error term in the model is not presumed to be constant over time. This approach was proposed by Engle (1982) to improve the modeling of financial and economic data but has proven invaluable in modeling any time-series variable in which there are periods of turbulence followed by a relative calm at some point. The estimates are obtained using the conditional maximum likelihood procedure. Please see Kennedy (2008, pp. 21-22, 124-125), Boffelli and Urga (2016, pp. 82-101), or Becketti (2013, pp. 271-285) for an overview of the method. Those interested in a more detailed explanation should consult Hamilton (1994, pp. 657-665) or Bollerslev (1986). One challenge associated with using the method is that model convergence can be difficult to achieve. In STATA's words, "ARCH models are notorious for having convergence difficulties. Unlike in most estimators in Stata, it is common for convergence to require many steps or even to fail. … This is particularly true of [ARCH-in-mean], and of any model with several lags in the ARCH terms. There is not always a solution." (STATA, 2021, p. 33). The view here is that the convergence challenge can be mitigated by attention to the structural drivers of the heteroskedasticity and the issue of the functional form of the exogenous inputs. For this reason, the text below contains commentary on these issues that may be helpful to future researchers. The second component of the time-series portion of the ARCH/ARMAX method is the autoregressive-moving-average (ARMA) method, which models the autocorrelations in hourly temperature based on autoregressive (AR) terms and moving average (MA) terms. A recent meteorological-related application of the ARMA method with exogenous inputs includes Forbes (2021) in an analysis of solar energy generation in Germany. A recent CO 2 -related application of the full ARCH/ARMAX method includes Forbes and Zampelli (2019) in an analysis of CO 2 emissions in the Irish power grid.
Including the model's exogenous inputs, an ARCH/ARMAX model can be represented using two equations. In the case of hourly data, the first equation models the conditional hourly mean, while the second models the conditional variance, which may include regressors to account for the structural component of the heteroskedasticity. Following Engle et al. (1987), the possible effect of the conditional variance on the conditional hourly mean, a phenomenon known as an ARCH-in-mean effect, can also be modeled. A possible example of the overall modeling approach when the natural logarithm of y t is the dependent variable, and there are J exogenous inputs in the condition hourly mean equation and K exogenous inputs in the conditional variance equation is where the X ′ j s represents the exogenous inputs in the conditional hourly mean equation, the Z ′ k s are the exogenous inputs in the conditional variance equation, g is the function (e.g., t−i instead of 2 t−i ) that reflects the conditional variance in Equation (1), Ψ i represents the estimated ARCH-in-mean effects for period t − i (where i could be equal to 0, 1, 2, 3, etc.), ∑ P AR(p) and ∑ Q MA(q) are the sums of the autoregressive and moving average terms, with p and q being the selective lags corresponding to the AR and MA terms, respectively. These equations are only illustrative. The various components of these equations are discussed in more detail below. In particular, much of the analysis will focus on whether the linear form of the X j 's is appropriate.
With Equations (1) and (2) in mind, the approach in this article accepts George Box's well-known aphorism that "All models are wrong; some models are useful" (Box et al., 2005, p. 440). They are all "wrong" in that all are simplifications of a complex reality, but they can be useful if they capture important aspects. A possible corollary of Professor Box's proposition is that a model can easily be portrayed as being terribly "wrong," even if it is useful. This point may help explain why Climate Deniers have so far been very successful in preventing the implementation of policies that substantially reduce CO 2 emissions, given that all they need to do is complain about the climate models that predict warming. For example, Lindzen has cast doubt by complaining about the representation of cloud cover in the climate models (Gillis, 2012).
One approach to the issue of model adequacy in time-series models follows Becketti (2013, p. 256), Granger (1986, p. 91), Kennedy (2008, and Granger and Newbold (1974, p. 119), who have stressed the importance of "white noise" in the residuals. The view here is that the issue of model adequacy should focus on usefulness and that the most important metric of a model's usefulness is its out-of-sample predictive accuracy, given that a "useless" model will, almost by definition, be incapable of making accurate out-of-sample predictions.
The modeling proceeds by discussing the exogenous inputs in Equation (1). It is hypothesized that the hourly CO 2 /temperature relationship may be highly conditional on forecasted temperature and forecasted humidity. The model employs CO 2 interaction terms to capture these possibilities. The focus on these specific interactions is based on the results of a less parsimonious version of the model that included all the possible interactions.
The linear form of the equation representing the exogenous inputs, that is, the structural portion of Equation (1), is Temp t = c + 1 FT t + 2 FP t + 3 FH t + 4 FDP t + 5 FWS t + 6 FV t where Temp t is the actual temperature in hour t, CO2 t−25 is the atmospheric CO 2 concentration level in hour t-25, FT t is the day-ahead forecasted temperature for hour t, FP t is the day-ahead forecasted probability of precipitation for hour t, FH t is the day-ahead forecasted humidity level for hour t, FDP t is the day-ahead forecasted dewpoint for hour t, FWS t is the day-ahead forecasted wind speed for hour t, FT t is the day-ahead forecasted visibility for hour t, FSKY t is a vector of binary variables representing forecasted sky conditions for hour t. The specific names of the binary variables are as follows: Cloudy, Sunny, ClearSky, MostlySunny, MostlyClear, Haze, ScatteredClouds, PartlyCloudy, HighLevelClouds, MoreSunthanClouds, MoreCloudsthanSun, PartlySunny, BrokenClouds, MostlyCloudy, and Overcast. To avoid the singularity associated with the well-known "dummy variable trap" (Kennedy, 2008, p. 506), Overcast is not explicitly recognized but is reflected in the overall constant term.
HR is a vector of binary variables representing the hour of the day in UT, exclusive of hour one, which is reflected in the constant term to avoid singularity.
Season. The word "season" is typically associated with spring, summer, fall, and winter. The view here is that seasonality is much more gradual in nature, with the last day of summer not being much different from the first day of fall. In this article, seasonality is modeled as a vector of binary variables representing each ten-day continuous period of the year, exclusive of January 1 through January 5, which is reflected in the constant term to avoid singularity. In 2012, a leap year, the binary variable that includes February 29 has 11 days.
Year k is a vector of binary variables for each year, exclusive of 2014. U t is a random error term for hour t.

ESTIMATION
As noted by Kennedy (2008, p. 102-103), a linear specification of the dependent variable is very convenient but can have adverse statistical consequences if the data do not support this specification. To avoid the possible shortcomings of assuming an inappropriate functional form, the estimation process proceeds by first assessing whether the dependent variable needs to be transformed to reduce its skewness. The specifications of the explanatory variables on the right-hand side of Equation (3) are also considered. With respect to the dependent variable, it is noted that while a Gaussian distribution has a skewness of zero, the data series representing temperature has a skewness of approximately 0.274. This skewness is reduced to approximately 0.015 by modeling the natural logarithm of temperature. Based on this result, the analysis focuses on modeling the natural logarithm of temperature. The specification of the exogenous inputs was explored using the multivariable fractional polynomial (MFP) model, a useful technique when one suspects that some or all of the relationships between the dependent variable and the explanatory variables are non-linear. As explained by Royston and Sauerbrei (2008), the MFP procedure begins by estimating a model that is strictly linear in the explanatory variables. Subsequent estimations cycle through a battery of nonlinear transformations of the explanatory variables (e.g., cube roots, square roots, squares, etc.) until the model that best predicts the dependent variable is found. In the present case, the set of exponents that the procedure considered include 0.25, 0.3333, 0.5, 0.6666, 0.75, 1, 1.5, 2, 2.5, and 3. In this case, the MFP analysis supported representing several explanatory variables with powers other than unity. For example, the MFP recommended exponent for the variable CFH t is three, while the recommended exponent for CFT t is one-half. The transformed form of the equation with the exogenous inputs is Please note that the terms in (4) denoted using the Greek alphabet (e.g., 1 ) now represent the estimated parameters obtained given this version of the model. Equation (4) is linear in terms of the transformed variables and thus is easily estimated using ordinary least squares. Least squares estimation of (4) yields an estimated equation with a seemingly respectable R 2 of about 0.8324. Unfortunately, the parameter estimates are statistically unreliable because a significant number of the autocorrelations in the residuals lie outside the 95% confidence band corresponding with the null hypothesis of no autocorrelation. The null hypothesis of no autoregressive conditional heteroskedasticity, that is, no ARCH effects, is rejected with a p-value less than 0.0001 using Engle's Lagrange multiplier test (Engle, 1982). Consistent with these issues, the predictions over the evaluation period have an RMSE of about 1.58 • C, higher than the error in CustomWeather's day-ahead temperature forecasts over the same period.
Fortunately, using ARCH/ARMA methods, the functional form presented in (4) can be used to generate much more accurate predictions than the predictions from a least-squares estimation of (4). It is recognized that some readers may be skeptical of the MFP-recommended transformations; thus, an ARCH/ARMAX model that does not employ those recommended transformations is also estimated.
The modeled lag lengths in the ARCH process are 1, 2, 3, 24, and 25. In addition, the structural component of the conditional variance, that is, the Zs in Equation (2), is modeled as a function of binary variables for the hour of the day, the year of the sample, and the following variables: and √ FT t . These variables are the square roots of the exogenous inputs in Equation (1). They are specified as square roots based on the superiority of the resulting AIC (Akaike, 1974) and BIC (Schwarz, 1978) values compared to when linear specifications were assumed. It is also worth noting that excluding these variables from the model resulted in an estimation process that did not converge. Specifically, the estimation process encountered a discontinuous region that made it impossible to compute an improvement in the objective function. One possible reason for this outcome is that hourly temperature is not independent of the conditional variance. As mentioned above in the discussion of Equation (1), the ARCH-in-mean model introduced by Engle et al. (1987) offers an approach to estimating this possible linkage. In this case, the modeled lags are 0, 1, and 2. For the AR(p) process, the modeled lag lengths are p = 1, 2, 3, 4, 5, 24, 25, 44, 45, 47, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, 408, 432, 624, 672, and 696. The moving-average nature of the disturbance terms is modeled for the lag lengths q = 1 through 24.

F I G U R E 5
The autocorrelations in the standardized residuals.

TA B L E 1
The ARCH/ARMAX estimation results when the CO 2 -related variables are included as covariates. The Portmanteau (Q) test was conducted for the hourly lags 1 through 800. For 50 of these 800 of these lags (lags 2 through 51), the p-values are less than 0.05, not supporting the null hypothesis of white noise. However, most of the autocorrelations in the standardized residuals are within the confidence bands corresponding to the null hypothesis of white noise ( Figure 5).

Variable
In contrast to the common practice of assuming that the error distribution is Gaussian, the estimation process employs the Student t distribution. The shape of this distribution approaches the Gaussian distribution when the degrees of freedom equals 30 or more. In this case, the estimated number of degrees of freedom is about 7.28, indicating that a nontrivial level of kurtosis characterizes the error distribution. Employing an algorithm reported by Harvey (2013, p. 20), the overall kurtosis level accommodated by this distribution in excess of the Gaussian distribution's level of three is approximately equal to two in this case. The advantage of using this distribution is that it helps ensures that the parameter estimates reflect the nature of the data.
The ARCH/ARMAX estimates based on the MFP transformations are reported in Table 1. The following three CO 2 -related estimated coefficients for the conditional mean equation are highly statistically significant: CO2 3 , CFT 1/2 , and CFH 3 . In the case of CFT 1/2 , the estimated coefficient is positive and nontrivial in magnitude, indicating that the estimated association between CO 2 concentrations and hourly temperature is higher, the higher the level of forecasted temperature.
Table 1 also indicates that the following continuous forecasted weather variables are statistically significant: FP 3/2 , FT 2/3 , FH, FDP, and FV 3 . Concerning the 14 forecasted sky variables, only the binary variable Cloudly is statistically significant. These forecasted sky results are possibly explained by the conceded low skill of the cloud cover forecasts (Haiden et al., 2015, p. 2). Twenty-one of the 23 binary variables representing the hours of the day are statistically significant. For the 36 binary variables representing the season of the year, 22 of the variables are statistically significant. For each year, these statistically significant terms span the entire period from April 16 to November 21. Four of the five binary variables representing each sample year are statistically significant.
Concerning the ARCH and ARCH-in-mean terms, all five ARCH terms are statistically significant, and two of the three ARCH-in-mean terms are statistically significant. The ARCH-in-mean terms in question correspond to lags one and two. Their positive and statistically significant values indicate that high levels of volatility in temperature are leading indicators of higher hourly temperatures. It is possible that this finding could contribute to an improved understanding of heat waves.
Concerning the ARMA terms, 28 of the 30 AR terms are statistically significant, and 24 of the 24 MA terms are statistically significant. Consistent with the autocorrelative nature of hourly temperature reported in Figure 4, these AR and MA terms provide predictive information concerning the actual temperature value in hour t, with the incremental information generally declining over time.
The structural component of the conditional variance is modeled using 38 variables exclusive of the constant term; 27 of these variables are statistically significant. Among the statistically significant variables are 15 of the 23 binary variables representing the hour of the day, four of the five variables representing the year of the sample, and , and √ FT t . Given these results, there are a total of seven CO 2 -related variables that have statistically relevant implications for temperature (three through the conditional mean equation and four via the conditional variance equation). Overall, the model has 191 variables. Eighty-eight of these variables are employed in the structural equation associated with the conditional mean equation, and 39 are used in the structural equation associated with the conditional variance equation. The remaining 64 variables either describe the error distribution or are time series in nature, the most prominent of these being the AR terms. Of the 191 variables, 145 are statistically significant. This relatively high incidence of statistical significance is encouraging, but it should be recognized that the true test of a model is its out-of-sample performance.

AN ALTERNATIVE MODEL THAT DOES NOT CONSIDER CO 2
This section considers an alternative model that does not consider CO 2 as a covariate. As before, the dependent variable is the natural logarithm of temperature. Applying the MFP procedure to ensure the best fit, the structural equation is ARCH/ARMAX and ARMAX models based on Equation (5) were estimated using the same time-series specifications and input data employed in the previous section, excluding the CO 2 -related variables. Models with a linear time-trend variable were also estimated. Table 2 reports the values corresponding to the AIC (Akaike, 1974) and the BIC (Schwarz, 1978) statistics for various modeling specifications. Observe that the AIC and BIC values are lower when the CO 2 -related variables are covariates in the modeling. The models with the linear time trend variable are especially interesting in this regard, given that the linear trend variable has a 0.816 correlation with CO2 t−25 but is nevertheless statistically insignificant in the model specifications that exclude the CO 2 -related covariates reported as statistically significant in Table 1.
Based on the AIC and BIC literature, as reported by Kennedy (2008, p. 105), Table 2 indicates that the formulations that include the CO 2 -related variables are generally the better specifications. The AIC and BIC values support using the ARCH/ARMAX specification relative to the simpler ARMAX approach, that is, the approach without the conditional variance equation. It is also worth noting that the AIC and BIC values support using the ARCH/ARMAX approach based on the MFP transformed covariates compared to assuming linearity in the covariates. Indicative of the importance of the MFP-recommended transformations, none of the CO 2 -related covariates are statistically significant when the recommendations are ignored.

THE OUT OF SAMPLE ANALYSIS OF THE ARCH/ARMAX MODEL THAT INCLUDE THE CO 2 -RELATED COVARIATES
The out-of-sample evaluation period is from January 1, 2015, through December 31, 2017. Over these 24,734 h, a persistence forecast has an RMSE of 1.05 • C. Two prediction series for the out-of-sample period were created using the structural estimates. The first prediction series is based on the estimated structural parameters and the data over the out-of-sample period. The second series is identical except that it assumes an average CO 2 concentration level of 278 ppm, the approximate concentration level recognized by the IPCC before industrialization (Ciais et al., 2013, p. 467). Both series are transformed back to their original units of measurement, considering that a naive detransformation has a potential bias (Granger & Newbold, 1976, pp. 196-197). This matter is addressed by employing a hybrid approach. The first component of this approach employs the debiasing factor for the natural logarithmic transformation recommended by Guerrero (1993, eq. 10). This method is useful but is limited in effectiveness because it presumes that the error distribution is Gaussian (Baum and Hurn, 2021, p.169), which is not the case here. The second component of the approach uses the results from the first component as input in a within-sample post-processing regression analysis without a constant term to estimate a scaling factor. This latter method is explained by Baum and Hurn (2021, p. 170). This hybrid approach yields an unbiased prediction of hourly temperature for the sample period. The method is applied to make the out-of-sample predictions, with the estimated scaling factor based exclusively on the sample data.
The model's out-of-sample structural temperature predictions have an RMSE of about 1.398 • C when the predictions use the CO 2 value lagged by 25 h as an input. In contrast, the structural temperature predictions have an RMSE of about 6.650 • C when the predictions use the preindustrial CO 2 value of 278 ppm. Visually, the differences between the two sets of prediction errors are substantial ( Figure 6). This indicates that the predicted temperature is lower when the preindustrial level of CO 2 is presumed, given that the errors are calculated as the actual minus predicted temperature, an approach F I G U R E 6 The errors in the out-of-sample structural predictions, January 1, 2015, through December 31, 2017.

F I G U R E 7
The first 350,000 autocorrelations in hourly temperature at MLO, January 1, 1977, through December 31, 2020 consistent with Hyndman and Athanasopoulos (2018, p. 3.3). Specifically, the mean of the structural temperature predictions is approximately equal to the mean actual temperature when the CO 2 value lagged by 25 h is used as an input but understates the actual temperature by about 6.297 • C when the preindustrial CO 2 value is employed.
The differences between the two scenarios are significantly muted when the full ARCH/ARMAX model is used to generate the predictions. This damping of the differences occurs because the predictions are affected by the ARCH/ARMA parameters, which keeps the hour-ahead predictions closely aligned with lagged temperature outcomes.
The full ARCH/ARMAX hour-ahead temperature predictions have an RMSE of about 1.124 • C when the CO 2 values are consistent with the preindustrial era; the temperature predictions have an RMSE of 0.728 • C when the CO 2 value lagged by 25 h is used as an input, that is, the prediction error is lower when the CO 2 levels from the modern era are used as inputs. While the prediction error is lower, the predicted temperature ranges from six degrees colder to about nine degrees warmer. On average, the mean predicted temperature is about 0.672 • C higher when the CO 2 levels from the modern era are used as inputs compared to when the preindustrial CO 2 levels are employed. This estimate is less than Craigmile and Guttorp's global estimate of 1.2 • C, but it is readily conceded that the 0.672 • C estimate understates the ultimate impact of higher CO 2 levels at the Hilo location because the autocorrelative process in hourly temperature can mute the effects for a considerable period of time. The autocorrelations in hourly temperature at MLO from January 1, 1977, through December 31, 2020, are informative in this regard (Figure 7). The local nature of the predicted value for Hilo and the methodology's reliance on forecasted weather inputs formulated in the modern era may also play a role in understanding the difference between the two estimates.

CONCLUSION
This analysis has examined the relationship between the hourly CO 2 atmospheric concentration level and hourly temperature. A CO 2 -augmented ARCH/ARMAX model was estimated using day-ahead weather forecast variables as controls.
The estimated associations between seven CO 2 -related variables and hourly temperature are highly statistically significant. More importantly, the out-of-sample evidence indicates that the magnitude of CO 2 's effect on hourly temperature is nontrivial. Accordingly, this article adds to the consilience of evidence in support of the scientific consensus. In one sense, the contribution is minor because it simply confirms what the vast proportion of climate scientists believe to be true. In another sense, the research justifies a climate perspective when considering the weather by rejecting the null hypothesis that CO 2 has trivial consequences for hourly temperature. This change may be important given the public's high interest in local weather conditions and the unsubstantiated claims by climate deniers that sharply dismiss the linkage between greenhouse gases and extreme weather events. Given that all models are "wrong," it is easy to dismiss the statistically significant results reported in Table 1. It is much more challenging to rationally dismiss the implications of the significant decline in the out-of-sample predictive accuracy when the preindustrial levels of CO 2 are assumed. One might claim that the out-of-sample findings result from a mere correlation between temperature and the hour t − 25 level of CO 2 over the out-of-sample period. The facts do not support this view: the simple correlation between the two variables over the out-of-sample evaluation period equals −0.1953.
One possible objection to the results presented in this article is that the model is "wrong." For example, it may be claimed that the parameter estimates are somehow biased because of the spatial nature of temperature, as reported in the Appendix. Objections of this nature are interesting approaches to debunking a model but have no merit. For as careful readers of this article will recognize, all models are "wrong," but the model presented here is useful because it isolates the CO 2 signal from the noisy data despite its spatial nature and then employs out-of-sample data to verify the findings.
It may be claimed that some unknown natural factor is the true culprit of the decline in predictive accuracy when the preindustrial levels of CO 2 are assumed. It is cheerfully conceded that this could be the case, that is, these results could result from the omission of a relevant explanatory variable. Such a variable must be highly correlated with the hourly CO 2 concentration level at MLO, and whose effects on hourly temperature are consistent with scientific principles. Those who believe the analysis presented here suffers from this possible deficiency should be able to name this variable instead of simply presuming its existence. As the philosopher of science Lipton (2004, p. 56) notes, "we cannot infer something simply because it is a possible explanation. It must somehow be the best of competing explanations." It is suspected that most individuals without any obscurantist agenda would agree that attributing the results presented in this article to completely unknown and possibly nonexistent factors is not the best of competing explanations.
Individuals tempted to conclude that the implications of the paper's findings are limited in scope to Hawaii are cheerfully invited to read the Appendix, which presents evidence that the temperature in Hawaii has predictive implications for the temperature at other locations. Considering the conclusions of Wynes (2022, p. 1404) and Washington D.C.'s highly challenging policy environment, the paper's findings suggest that the current outlook for the Earth's future is quite grim. However, the public's high interest in hourly weather conditions and the results presented here that strongly indicate that CO 2 affects those conditions might help induce a new era in terms of public attitudes and policies. One possible research path that might enhance the payoff from this more weather-centric policy environment is a demonstration project that would report near-real-time hourly temperature predictions under the assumption of both near-current CO 2 levels and the levels that prevailed before industrialization. At a minimum, the project would make the public more aware of the consequences of climate policy inaction.

ACKNOWLEDGMENTS
The paper's results rely on the data collected at the Mauna Loa Observatory. I thank the Earth System Research Laboratory (ESRL), Global Monitoring Division (GMD) of the National Oceanic and Atmospheric Administration (NOAA) for supporting the operations of this observatory. I am particularly grateful to Dr. Pieter Tans for alerting me to NOAA ESRL's quality control procedures. I thank NOAA's National Climatic Data Center for its data. I thank CustomWeather for providing me with the forecast data and permitting its dissemination. I am particularly thankful to Richard Reed, Thomas Hauf, and Agustin Diaz for answering my many questions about the data. I thank Met Éireann, the Irish National Meteorological Service, for making the hourly historical weather data available for its Valentia Observatory. I also thank CustomWeather for providing me with the hourly temperature data for Central Park in New York City in the United States, Beijing International Airport in China, and the other airport locations. I thank Michael A. Forbes for writing a script to download the hourly day-ahead forecasts for the Hilo location reported by the National Weather Service. I benefited from discussions with Seán Lyons. I have also benefited from Roger Vadim and Stephan Kolassa's modeling advice. I also found the comments by two anonymous reviewers to be invaluable. The views expressed in this article are those of the author and do not necessarily reflect those of the individuals and organizations listed above. Any errors are the full responsibility of the author.

FUNDING INFORMATION
This publication has emanated from research conducted at The Catholic University of America in Washington DC, USA, the Economic and Social Research Institute in Dublin, Ireland, and University College Dublin in Ireland. It was partly supported by a Grant from Science Foundation Ireland under Grant number 15/SPP/E3125 but was largely self-funded.

CONFLICT OF INTEREST STATEMENT
The author is recognized as a contributor to two patents that address the challenges of integrating renewable energy into the power grid. Information about the patents is available at the following links: https://patentscope.wipo. int/search/en/detail.jsf;jsessionid=F21BECE874FB500486D2F6CA4C321F02.wapp2nB?docId=WO2017201427&tab= PCTDESCRIPTION; https://patentscope.wipo.int/search/en/detail.jsf?docId=US339099329&docAn=17230201

DATA AVAILABILITY STATEMENT
The data that supports the findings of this study are available in the supplementary material of this article ORCID Kevin F. Forbes https://orcid.org/0000-0002-9521-6845