A Spatial Econometric Analysis of Weather Effects on Milk Production

Abstract

Greenhouse gas (GHG) emission-induced climate change, particularly occurring since the mid-20th century, has been considerably affecting short-term weather conditions, such as increasing weather variability and the incidence of extreme weather-related events. Milk production is sensitive to such changes. In this study, we use spatial panel econometric models, the spatial error model (SEM) and the spatial Durbin model (SDM), with a panel dataset at the state-level varying over seasons, to estimate the relationship between weather indicators and milk productivity, in an effort to reduce the bias of omitted climatic variables that can be time varying and spatially correlated and cannot be directly captured by conventional panel data models. We find an inverse U-shaped effect of summer heat stress on milk production per cow (MPC), indicating that milk production reacts positively to a low-level increase in summer heat stress, and then MPC declines as heat stress continues increasing beyond a threshold value of 72. Additionally, fall precipitation exhibits an inverse U-shaped effect on MPC, showing that milk yield increases at a decreasing rate until fall precipitation rises to 14 inches, and then over that threshold, milk yield declines at an increasing rate. We also find that, relative to conventional panel data models, spatial panel econometric models could improve prediction performance by leading to smaller in-sample and out-sample root mean squared errors. Our study contributes to the literature by exploring the feasibility of promising spatial panel models and resulting in estimating weather influences on milk productivity with high model predicting performance.

1. Introduction

The increasing concentration of greenhouse gases (GHGs) in the Earth’s atmosphere is expected to increase air temperatures, weather variability, and the frequency of weather-related extreme events such as heat waves and freezes [1]. The literature indicates milk production, feed intake, reproductive efficiency, and animal health are all sensitive to such changes [2]. Regional effects are expected to be uneven. Understanding the impacts of weather condition changes on regional milk production in the U.S. is crucial for the U.S. agricultural sector and is attracting much attention from researchers [3,4]. In addition, understanding weather influences on agricultural production can provide practical insights for farmers to implement adequate adaptation and mitigation strategies to achieve the goals of balancing economic and environmental sustainability.

In this study, we use panel econometric models to estimate relationships between weather indicators and milk productivity. Panel data approaches have been widely utilized to analyze climatic effects on productivity, as reviewed in Blanc and Schlenker [5]. However, such approaches may suffer from omitted variable biases [6]. In particular, time-varying omitted variables that are common across regions cannot be captured by regional fixed effects. Weather factors, such as wind or solar radiation, vary over time and are spatially correlated. When omitted, such variables will lead to spatially correlated error terms. Spatial panel models have been developed to handle such issues [7,8,9,10]. Most recent studies that apply spatial panel models usually focus on crop production [11,12], or breeding quantity of livestock [13], or environmental factors that can be correlated with livestock [14,15]. To the best of our knowledge, we are not aware of any study that has applied spatial panel models to examine the extent to which changes in weather conditions can affect milk production to reduce omitted variable biases. The rest of the paper is organized as follows. Section 2 reviews related literature. The model structure, assumptions, and data are described in Section 3 and are followed by the model estimation results in Section 4. Conclusions and discussion are provided in Section 5.

2. Literature Review

2.1. Weather and Milk Productivity

Heat stress and the zone of thermal comfort play an important role in livestock production and reproduction [16]. When conditions exceed the upper limit of that zone, livestock exhibit degraded performance [17]. West [18] and Lacetera et al. [19] indicate dairy cows are especially sensitive to heat stress, exhibiting reduced feed intake, increased water intake, decreased meat and milk production, and altered birth rates. Ravagnolo, Misztal, and Hoogenboom [20] suggest that a one-unit increase in heat stress leads to about a 0.2 kg daily milk yield loss compared to a base of 26.3 kg. Mauger et al. [4] estimate climate is causing present-day US Holstein milk production losses of 1.9 percent and project that climate change could cause this to increase to 6.3 percent by 2100, generating a USD 2.2 billion annual loss.

Heat stress arises from the combination of temperature, air movement, and humidity [16,21,22,23]. The temperature–humidity index (THI) is the most commonly used index [24,25] to quantify heat stress. There are several alternative THI formulas, as summarized by Bohmanova et al. [21]. The determinants are dew point temperature, wet bulb temperature, relative humidity, wind speed, solar radiation, and water vapor [26]. The study by Amundson et al. [27] calculates the THI using air temperature and relative humidity and finds that the THI threshold is about 73 and the pregnancy rate in beef cattle will decline when the THI exceeds that threshold. Hammami et al. [28] show milk production exhibits an inverse U-shaped curve with respect to THI, with the threshold falling in the range of 62–82. In particular, milk productivity is expected to increase at a decreasing rate when the THI increases before reaching a threshold level, and then milk yield decreases at an increasing rate as the THI continues increasing.

2.2. Econometric Analysis of Climate Effects on Agricultural Production

There have been a number of panel econometric studies measuring weather impacts on agricultural productivity, as reviewed in Blanc and Schlenker [5]. For crop yields, such studies include: Schlenker and Roberts [29], Deschênes and Greenstone [30], Miao, Khanna, and Huang [31], etc. Regionally, Mukherjee, Bravo-Ureta, and De Vries [32] examine climatic impacts on southeastern U.S. milk yields, finding that THI has a significant non-linear negative effect. Qi, Bravo-Ureta, and Cabrera [33] examine temperature and precipitation effects on Wisconsin dairy farm productivity and find those weather indicators have exerted deleterious effects and would further reduce milk output by 5 to 11% per year between 2020 and 2039. Perez-Mendez, Roibas, and Wall [34] estimate the effects of weather conditions on Spanish milk production, considering cow performance and forage supplies. But they find changes in weather mainly affect forage availability, and the THI has no significant effect on milk production. Key and Sneeringer [35] examine heat stress in US major milk-producing regions, finding it would depress milk production and cause national economic loss.

Although panel models control some forms of omitted variable bias [5,36], they do not control all forms. In particular, omitting climatic variables that are related across regions (e.g., vapor pressure deficit, solar radiation, and wind speed) would bias estimation results [37,38]. This is a concern since climatic variables inherently vary across time and are correlated across space [39]. Auffhammer et al. [40] argue that omission of such variables will cause spatially correlated residuals, which will then bias estimation results, and that the situation can be improved by using spatial panel models. Moreover, those weather variables omitted can be correlated with some existing weather variables included in the modeling analysis. For example, wind speed that is spatially dependent can affect heat stress [41,42] and thereby the extent of the heat stress influence on milk production across locations, which also necessitates the need of applying spatial econometric models.

Several studies have employed spatial panel models in studying climatic effects on crop yields. Schlenker, Hanemann, and Fisher [9] use the spatial error model (SEM), assuming spatially dependent error terms that are uncorrelated with independent variables. Ortiz-Bobea [10] uses the spatial Durbin model (SDM) that assumes spatially correlated error terms that are also correlated with independent variables. Herein, this study utilizes both SEM and SDM, analyzes their estimation results, and compares model performance with conventional panel data models to explore the feasibility of spatial panel models.

3. Materials and Methods

3.1. Data

The above non-spatial and spatial panel models are estimated with the dependent variable being annual milk production per cow (MPC), and the explanatory variables being state-level seasonal values of the palmer drought severity index (PDSI), total precipitation, and maximum and minimum THI. We use a dataset from 1950 to 2022 for the 48 contiguous US states. This yields 3504 observations and a balanced panel. Statistics computed over the data are shown in

Table 1. Statistical summary over historical data (1950–2022).

Variable	Obs. Count	Mean	Std. Dev.	Min.	1.Quartile	2.Quartile	3.Quartile	Max.
Log Milk Production per Cow	3504	4.1	0.2	3.4	3.9	4.1	4.2	4.4
Spring Precipitation	3504	9.7	4.4	0.3	6.6	9.6	12.5	31.1
Spring PDSI	3504	0.1	2.2	−7.45	−1.43	0.0	1.7	7.7
Spring Min THI	3504	40.1	7.8	23.5	34.0	39.3	45.5	63.3
Spring Max THI	3504	60.7	5.7	46.7	56.2	60.0	64.8	76.7
Summer Precipitation	3504	10.3	4.9	0.1	6.9	10.7	13.4	28.8
Summer PDSI	3504	0.1	2.4	−8.68	−1.53	0.2	1.9	9.1
Summer Min THI	3504	59.3	6.3	44.6	54.6	59.2	64.1	73.9
Summer Max THI	3504	74.7	4.2	64.8	71.2	74.5	77.7	89.1
Fall Precipitation	3504	8.8	4.3	0.5	5.3	8.8	11.7	24.5
Fall PDSI	3504	0.3	2.4	−7.92	−1.34	0.2	1.9	10.5
Fall Min THI	3504	43.0	7.1	26.5	37.6	42.5	47.9	66.2
Fall Max THI	3504	62.5	5.4	47.7	58.2	62.1	66.4	78.5
Winter Precipitation	3504	8.2	5.0	0.5	3.7	8.1	11.4	27.6
Winter PDSI	3504	0.2	2.0	−6.57	−1.09	0.2	1.5	6.8
Winter Min THI	3504	23.8	9.9	−3.47	16.9	23.8	30.3	53.1
Winter Max THI	3504	46.5	8.5	22.5	40.5	46.0	53.2	69.3

Note: PDSI denotes palmer drought severity index; THI denotes temperature humidity index.

.

Data sources and manipulations are as follows:

Data on annual milk production per cow are drawn from the United States Department of Agriculture’s National Agricultural Statistics Service Quick Stats database (please refer to https://quickstats.nass.usda.gov/, accessed on 20 June 2023) and are measured in pounds per cow in a year (lbs/head). State level daily palmer drought severity index (PDSI), total precipitation, minimum and maximum temperature, and minimum and maximum relative humidity (RH) data are drawn from the National Oceanic and Atmospheric Administration (NOAA) (please refer to https://www.ncei.noaa.gov/cdo-web/, accessed on 20 June 2023). These data are aggregated into 4 seasons (spring-March to May, summer-June to August, fall-September to November, and winter-December to February). Over these periods, PDSI is the average value, and the more negative the value, the drier the climate is, with positive values being wetter; precipitation data are the three-month total. Daily minimum temperature and maximum RH are used to calculate daily minimum THI (min THI), whereas daily maximum THI (max THI) are calculated using maximum temperature and minimum RH [43,44], following the THI formula of Amundson et al. [27] and Mader, Davis, and Brown-Brandl [22]. Seasonal min THI and max THI are computed by averaging the values across the days in the three-month period. Temperatures are measured in degrees Celsius, and precipitation is measured in inches. We use the data from 1950–2014 to execute model estimation and reserve data from 2015–2022 for out-of-sample testing.

In our estimation, quadratic terms for the weather variables are included, following previous studies [45,46,47], as milk yield tends to be non-linearly correlated with a given weather variable. A linear and a quadratic time trend are also added as independent variables to capture “the effect of technological progress”. The dependent variable—MPC—is transformed into log form (as recommended by Mauldon [48] and Mosheim [49]), which helps stabilize the variance in the data [50].

3.2. Model Specification

A standard panel model [51,52] can be expressed as follows:

$$y=X\beta +\mu$$

(1)

where in our case $y$ is annual milk production per cow; $X$ is a vector of independent variables (i.e., weather indicators in our study); $\mu$ is an idiosyncratic error that is assumed to be independently identically distributed (iid).

We use two modeling approaches that improve estimations by controlling spatial correlation that arises due to omitted variables. The first is the SEM model of LeSage [53]. SEM assumes the omitted covariates are orthogonal to the independent variables but exhibit spatial dependence. That is in the above model $\mu =\gamma W\mu +\epsilon$ or equivalently,

$${\mu =(I_n-\gamma W)}^{-1}\epsilon$$

(2)

where $\gamma$ is a spatial dependence parameter; $I_n$ is an identity matrix; $W$ is a spatial weight matrix that reflects interactions in errors between regions; and $\epsilon$ is an iid error term. In this approach, Equation (1) is changed to be:

$$y=X\beta +{(I_n-\gamma W)}^{-1}\epsilon$$

(3)

Second is the SDM model of Cook, Hays, and Franzese [54]. Therein the omitted variables are assumed to be correlated with the independent variables. In this case, we assume the error term in (3) is given by

$$\epsilon =X\xi +v$$

(4)

where $\xi$ is the interrelationship coefficient; and $v$ is an iid error term. Then, inserting (4) into (3), we obtain:

$$y=\gamma Wy+X(\beta +\xi )+WX(-\gamma \beta )+v$$

(5)

For simplicity of notation, let

$$\tau =\beta +\xi ,\phi =-\gamma \beta$$

(6)

resulting in

$$y=\gamma Wy+X\tau +WX\phi +v$$

(7)

Equation (7) is the expression of the SDM estimating model and includes both the spatially lagged dependent variable and the independent variables. When $\xi =0$ in Equation (4), the SDM reduces to the SEM.

Following Belotti, Hughes, and Piano Mortari [55], a unifying specification encompassing both cases is:

$$y_{nt}=\alpha +\rho W{y}_{nt}+X_{nt}\beta +W{X}_{nt}\theta +{\varphi }_n+c_t+{\mu }_{nt}$$

(8)

$${\mu }_{nt}=\lambda W{\mu }_{nt}+{\epsilon }_{nt}$$

where $W$ is a spatial weight matrix; $\rho ,\theta ,\lambda$ are spatial correlation parameters; ${\varphi }_n$ and $c_t$ are the individual- and time-specific effects; $u_{nt}$ is a spatially correlated error; ${\epsilon }_{nt}$ is an idiosyncratic error.

In (8), if $\rho$ and $\theta$ are all equal to zero, the model reduces to the SEM case with the spatial interaction in the error term as denoted by $\lambda W{\mu }_{nt}$, which means that milk production in one region could be affected by unobserved factors in neighboring regions.

On the other hand, if $\lambda =0$, the model reduces to the SDM case, and $\beta$ and $\theta$ are coefficients expressing the direct and indirect (spatial spillover) effects of the independent variables. Finally, if $\rho =\lambda =\theta =0$, this reduces to a common panel model without spatial interaction.

In the spatial models, the matrix $W$ reflects regional proximity. In this case, we follow Fischer and Getis [56] and set the elements to $1/n$ for each of n neighboring states and zero for non-neighboring states. We also follow Taha et al. [57] and use the queen criterion to identify neighboring states.

4. Results

Estimations are done using conventional and spatial panel regression models. We estimate a pooled OLS, a fixed effects panel, and spatial panel models (i.e., SEM and SDM).

4.1. Results without Spatial Interactions

Pooled OLS and fixed effects panel model results are summarized in

Table 2. Pooled OLS and fixed effects model results.

Variable	Pooled OLS		Fixed Effects Model
Spring Precipitation	−0.0060 ***	−0.0012	−0.0018	−0.0011
Spring Precipitation 2	0.0001 **	0.0000	0.0001 **	0.0000
Spring PDSI	0.0033 **	−0.0014	−0.0020 *	−0.0011
Spring PDSI 2	0.0005 *	−0.0003	0.0002	−0.0002
Spring Min THI	0.0247 ***	−0.0041	0.0028	−0.0036
Spring Min THI 2	−0.0003 ***	−0.0001	−0.0001	0.0000
Spring Max THI	−0.0581 ***	−0.011	−0.0322 ***	−0.0091
Spring Max THI 2	0.0004 ***	−0.0001	0.0003 ***	−0.0001
Summer Precipitation	−0.0121 ***	−0.0011	−0.0016	−0.0014
Summer Precipitation 2	0.0004 ***	0.0000	0.0000	0.0000
Summer PDSI	−0.0002	−0.0012	0.0009	−0.001
Summer PDSI 2	0.0001	−0.0002	0.0005 ***	−0.0002
Summer Min THI	0.0412 ***	−0.0068	−0.0289 ***	−0.0084
Summer Min THI 2	−0.0004 ***	−0.0001	0.0002 ***	−0.0001
Summer Max THI	−0.0017	−0.0184	0.0476 ***	−0.017
Summer Max THI 2	−0.0001	−0.0001	−0.0003 ***	−0.0001
Fall Precipitation	0.0058 ***	−0.0014	0.0030 **	−0.0013
Fall Precipitation 2	−0.0002 ***	−0.0001	−0.0001 **	0.0000
Fall PDSI	−0.0037 ***	−0.001	−0.0014	−0.0009
Fall PDSI 2	−0.0003 *	−0.0002	−0.0004 **	−0.0002
Fall Min THI	0.0163 ***	−0.0037	−0.0167 ***	−0.0037
Fall Min THI 2	−0.0001 **	0.0000	0.0002 ***	0.0000
Fall Max THI	−0.0155	−0.01	0.0403 ***	−0.0087
Fall Max THI 2	0.0001	−0.0001	−0.0003 ***	−0.0001
Winter Precipitation	−0.0025 ***	−0.0009	−0.0004	−0.0009
Winter Precipitation 2	0.0000	0.0000	0.0000	0.0000
Winter PDSI	0.0005	−0.0013	0.0016	−0.001
Winter PDSI 2	−0.0008 **	−0.0003	−0.0006 **	−0.0002
Winter Min THI	−0.0030 ***	−0.0012	−0.0039 ***	−0.0011
Winter Min THI 2	0.0001 ***	0.0000	0.0001 ***	0.0000
Winter Max THI	0.0130 ***	−0.0035	0.0153 ***	−0.0033
Winter Max THI 2	−0.0001 ***	0.0000	−0.0002 ***	0.0000
T	0.0152 ***	−0.0002	0.0157 ***	−0.0002
T 2	−0.0001 ***	0.0000	−0.0001 ***	0.0000
Constant	4.3068 ***	−0.5264	2.6282 ***	−0.5308
R−squared	0.9342		0.9587

Note: PDSI denotes palmer drought severity index; THI denotes temperature humidity index; variable ending with superscript 2 denotes the quadratic form of a given variable; the asterisks represent the probability that the coefficient differs from 0 and * notes significance at 10% level, ** at 5%, and *** at 1%.

. Therein most weather variables exhibit a U-shaped effect with a significant quadratic term. The fixed effects model fits better as the null hypothesis of no state-specific fixed effects is rejected at the 1% level using the standard F test. The climate effects in the pooled OLS model are generally larger, indicating neglecting regional fixed effects overstates the climate impact. The smaller results under the panel model likely occur since its fixed effects terms capture important regional omitted variables such as soils, elevation, topography, etc.

In the fixed effects panel model, the summer max THI exhibits an inverse U-shaped effect, meaning that holding other factors constant, MPC will increase as the THI value increases up to a threshold value, and then if the summer max THI increases beyond that threshold level, MPC will decrease. This is generally consistent with findings from previous research [58].

4.2. Spatial Panel Model Results

The spatial panel models are relevant when there is spatial dependence in the error terms. We test this using Pesaran’s CD test [59,60] computed over the fixed effects panel model residuals. As a result, the no cross-regional dependence null hypothesis is rejected at the 1% level. Hence, it is appropriate to move on to estimation results from the spatial panel models.

The SEM and SDM estimation results are shown in

Table 3. SEM and SDM estimation results.

Variable	SEM		SDM
	Main		Main		WX
Spring Precipitation	−0.0024 **	−0.001	−0.0021 *	−0.0011	0.0026	−0.0017
Spring Precipitation 2	0.0001 ***	0.0000	0.0001 **	0.0000	−0.0001	−0.0001
Spring PDSI	−0.0005	−0.001	−0.0002	−0.001	−0.001	−0.0014
Spring PDSI 2	−0.0001	−0.0002	−0.0002	−0.0002	0.0006 **	−0.0003
Spring Min THI	0.0047	−0.0039	0.0066	−0.0044	−0.0088	−0.0062
Spring Min THI 2	−0.0001 **	0.0000	−0.0001 **	−0.0001	0.0001 *	−0.0001
Spring Max THI	−0.0242 **	−0.0094	−0.0165	−0.0103	0.0109	−0.0154
Spring Max THI 2	0.0002 ***	−0.0001	0.0002 *	−0.0001	−0.0001	−0.0001
Summer Precipitation	−0.0022 *	−0.0012	−0.0017	−0.0012	0.0039 *	−0.002
Summer Precipitation 2	0.0001 *	0.0000	0.0001	0.0000	−0.0002 **	−0.0001
Summer PDSI	0.0005	−0.0009	0.0004	−0.0009	0.0000	−0.0013
Summer PDSI 2	0.0003 **	−0.0001	0.0004 **	−0.0001	−0.0002	−0.0002
Summer Min THI	0.0242 ***	−0.0084	0.0352 ***	−0.009	−0.0812 ***	−0.0131
Summer Min THI 2	−0.0002 ***	−0.0001	−0.0003 ***	−0.0001	0.0007 ***	−0.0001
Summer Max THI	−0.0447 ***	−0.0158	−0.0474 ***	−0.0165	0.1444 ***	−0.025
Summer Max THI 2	0.0003 ***	−0.0001	0.0003 ***	−0.0001	−0.0010 ***	−0.0002
Fall Precipitation	0.0001	−0.0012	−0.0004	−0.0012	0.0036 *	−0.0018
Fall Precipitation 2	0.0000	0.0000	0.0000	0.0000	−0.0001 **	−0.0001
Fall PDSI	0.0008	−0.0008	0.0012	−0.0008	−0.0033 ***	−0.0012
Fall PDSI 2	−0.0004 **	−0.0001	−0.0003 **	−0.0001	0.0004 *	−0.0002
Fall Min THI	−0.0029	−0.0041	0.0041	−0.0046	−0.0164 ***	−0.0062
Fall Min THI 2	0.0000	0.0000	−0.0001	−0.0001	0.0002 ***	−0.0001
Fall Max THI	0.0124	−0.0094	0.0065	−0.0106	0.0228	−0.0149
Fall Max THI 2	−0.0001	−0.0001	−0.0001	−0.0001	−0.0002	−0.0001
Winter Precipitation	−0.0015	−0.001	−0.0020 **	−0.001	0.0015	−0.0014
Winter Precipitation 2	0.0000	0.0000	0.0000	0.0000	0.0000	−0.0001
Winter PDSI	−0.0002	−0.001	−0.0005	−0.001	0.0018	−0.0014
Winter PDSI 2	0.0001	−0.0002	0.0002	−0.0002	−0.0010 ***	−0.0003
Winter Min THI	−0.0009	−0.0013	0.0013	−0.0015	−0.0039 *	−0.002
Winter Min THI 2	0.0001 **	0.0000	0.0000	0.0000	0.0001	0.0000
Winter Max THI	0.0087 **	−0.0037	0.0047	−0.0043	0.0037	−0.006
Winter Max THI 2	−0.0001 ***	0.0000	−0.0001	0.0000	0.0000	−0.0001
T	0.0155 ***	−0.0003	0.0059 ***	−0.0003
T 2	−0.0001 ***	0.0000	−0.0000 ***	0.0000
λ	0.6651 ***	−0.0142
ρ			0.6239 ***	−0.0148
R−squared	0.8932		0.8818

Note: PDSI denotes palmer drought severity index; THI denotes temperature humidity index; variable ending with superscript 2 denotes the quadratic form of a given variable; the asterisks represent the probability that the coefficient differs from 0 and * notes significance at 10% level, ** at 5%, and *** at 1%.

. For the SEM, the spatial parameter λ is positive (0.67) and significant at the 1% level, implying a strong spatial dependence in the unobserved factors. The effects of a random shock in a specific region that transmit to its neighborhood can be estimated by multiplying λ with the corresponding spatial weights.

Now turning to the SDM results, the spatial autoregressive parameter ρ (0.62), which reflects the average intensity of the spatial intercorrelation [53], is also significant, indicating a strong spatial dependence. The spatially lagged independent variables ($WX$) measure the indirect impacts on MPC in one region arising from changes in variables $X$ from its neighboring regions [61]. Following LeSage and Dominguez [62] and Liu and Nie [63], weather influences can be split up considering the direct in-region effects of $X$ and the indirect effects translated in from adjacent regions. In particular, Equation (7) can be rewritten to Equation (9):

$$y_{nt}={\sum }_{k=1}^K{\left(I_n-\gamma W\right)}^{-1}\left(I_n{\tau }_k+W{\phi }_k\right)x_{k,nt}+{\left(I_n-\gamma W\right)}^{-1}{\nu }_{nt}$$

(9)

and the effects of the independent variables can be computed as

$$S_k={\displaystyle \frac{\partial y_{nt}}{\partial x_{k,nt}}}={\left(I_n-\gamma W\right)}^{-1}\left(I_n{\tau }_k+W{\phi }_k\right)={\left(I_n-\gamma W\right)}^{-1}\left[\begin{array}{ccc}\begin{array}{cc}{\tau }_k& w_{12}{\phi }_k\\ w_{21}{\phi }_k& {\tau }_k\end{array}& \cdots & \begin{array}{c}{w}_{1n}{\phi }_k\\ w_{2n}{\phi }_k\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{cc}{w}_{n1}{\phi }_k& w_{n2}{\phi }_k\end{array}& \cdots & {\tau }_k\end{array}\right]$$

where $n$ identifies region, $k=1,\dots ,K$ denotes the set of independent variables, and $S_k$ the effect on production across regions of variations in the $k^{th}$ independent variable. In turn, the average direct effect is the average of the diagonal elements of $S_k$, and the average total effect is equal to $n^{-1}{\iota }_n^{\text{'}}{S}_k{\iota }_n$, which ${\iota }_n$ is a vector of 1’s. The indirect effect is the total effect minus the direct effect.

For our estimation, the empirical SDM direct and indirect impacts are shown in

Table 4. Direct and indirect effects of climate variables from SDM (vs. SEM results).

Variable	SDM						SEM
	Direct Effect		Indirect Effect		Total Effect
Spring Precipitation	−0.0018 *	−0.0011	0.0030	−0.0032	0.0012	−0.0034	−0.0024 **	−0.001
Spring Precipitation 2	0.0001 *	0.0000	−0.0001	−0.0001	0.0000	−0.0001	0.0001 ***	0.0000
Spring PDSI	−0.0004	−0.0009	−0.0029	−0.003	−0.0033	−0.0033	−0.0005	−0.001
Spring PDSI 2	−0.0001	−0.0002	0.0012 *	−0.0006	0.0010	−0.0007	−0.0001	−0.0002
Spring Min THI	0.0056	−0.0039	−0.0115	−0.0111	−0.0059	−0.0109	0.0047	−0.0039
Spring Min THI 2	−0.0001 **	0.0000	0.0002	−0.0001	0.0001	−0.0001	−0.0001 **	0.0000
Spring Max THI	−0.0164 *	−0.0094	0.0007	−0.0284	−0.0157	−0.028	−0.0242 **	−0.0094
Spring Max THI 2	0.0002 **	−0.0001	0.0000	−0.0002	0.0001	−0.0002	0.0002 ***	−0.0001
Summer Precipitation	−0.001	−0.0012	0.0067	−0.0042	0.0057	−0.0046	−0.0022 *	−0.0012
Summer Precipitation 2	0.0000	0.0000	−0.0003 **	−0.0001	−0.0003 *	−0.0002	0.0001 *	0.0000
Summer PDSI	0.0003	−0.0008	0.0008	−0.0026	0.0011	−0.0029	0.0005	−0.0009
Summer PDSI 2	0.0004 **	−0.0002	0.0001	−0.0005	0.0005	−0.0006	0.0003 **	−0.0001
Summer Min THI	0.0210 **	−0.0085	−0.1449 ***	−0.0238	−0.1239 ***	−0.0254	0.0242 ***	−0.0084
Summer Min THI 2	−0.0002 ***	−0.0001	0.0012 ***	−0.0002	0.0010 ***	−0.0002	−0.0002 ***	−0.0001
Summer Max THI	−0.0213	−0.0159	0.2808 ***	−0.0485	0.2594 ***	−0.0543	−0.0447 ***	−0.0158
Summer Max THI 2	0.0001	−0.0001	−0.0019 ***	−0.0003	−0.0018 ***	−0.0004	0.0003 ***	−0.0001
Fall Precipitation	0.0004	−0.0012	0.0080 **	−0.0036	0.0084 **	−0.0038	0.0001	−0.0012
Fall Precipitation 2	0.0000	0.0000	−0.0003 **	−0.0001	−0.0003 **	−0.0001	0.0000	0.0000
Fall PDSI	0.0006	−0.0008	−0.0063 **	−0.0025	−0.0056 **	−0.0027	0.0008	−0.0008
Fall PDSI 2	−0.0003 **	−0.0001	0.0003	−0.0004	0.0000	−0.0005	−0.0004 **	−0.0001
Fall Min THI	0.0007	−0.0044	−0.0336 ***	−0.0106	−0.0328 ***	−0.0103	−0.0029	−0.0041
Fall Min THI 2	0.0000	−0.0001	0.0004 ***	−0.0001	0.0004 ***	−0.0001	0.0000	0.0000
Fall Max THI	0.0128	−0.0098	0.0658 **	−0.0258	0.0785 ***	−0.0251	0.0124	−0.0094
Fall Max THI 2	−0.0001	−0.0001	−0.0005 ***	−0.0002	−0.0007 ***	−0.0002	−0.0001	−0.0001
Winter Precipitation	−0.0020 **	−0.001	0.0005	−0.0027	−0.0015	−0.0028	−0.0015	−0.001
Winter Precipitation 2	0.0000	0.0000	0.0001	−0.0001	0.0001	−0.0001	0.0000	0.0000
Winter PDSI	−0.0001	−0.001	0.0038	−0.0029	0.0037	−0.0032	−0.0002	−0.001
Winter PDSI 2	0.0000	−0.0002	−0.0020 ***	−0.0007	−0.0019 ***	−0.0007	0.0001	−0.0002
Winter Min THI	0.0006	−0.0014	−0.0076 **	−0.0034	−0.0070 **	−0.0033	−0.0009	−0.0013
Winter Min THI 2	0.0000	0.0000	0.0001 *	−0.0001	0.0002 **	−0.0001	0.0001 **	0.0000
Winter Max THI	0.0062	−0.0038	0.0163 *	−0.0098	0.0225 **	−0.0094	0.0087 **	−0.0037
Winter Max THI 2	−0.0001 **	0.0000	−0.0002	−0.0001	−0.0003 **	−0.0001	−0.0001 ***	0.0000
T	0.0067 ***	−0.0002	0.0089 ***	−0.0003	0.0156 ***	−0.0003	0.0155 ***	−0.0003
T 2	−0.0000 ***	0.0000	−0.0001 ***	0.0000	−0.0001 ***	0.0000	−0.0001 ***	0.0000

Note: PDSI denotes palmer drought severity index; THI denotes temperature humidity index; variable ending with superscript 2 denotes the quadratic form of a given variable; the asterisks represent the probability that the coefficient differs from 0 and * notes significance at 10% level, ** at 5%, and *** at 1%.

, along with the SEM results for comparison. Overall, the SDM estimates show that more weather variables exert significant effects relative to the SEM results. Summer max THI again exhibits an inverse U-shape total effect on MPC; thereby milk yield increases as the summer max THI increases toward a threshold value, and then yield declines as this THI is higher than the threshold moving extreme heat stress, which coincides with the findings in Hammami et al. [28]. We also find a significant indirect effect of summer max THI, which represents the spatial spillover impacts on MPC in a state from changes in summer max THI of its nearby states [10].

Additionally, the linear and quadratic time trend variables were included here to capture the effects of factors like technological improvements changing over time, such as developments in animal genetics, nutrition, and animal management. In both spatial models and the pooled OLS and non-spatial fixed effects panel model, the linear time trend is significant and positive, showing increasing MPC over time, while the quadratic term is negative, showing diminishing technological progress over time.

These results can be illustrated graphically. Figure 1 displays the relationship between a given weather factor and the MPC when holding other variables constant. Figure 1a shows the percentage changes in MPC as the summer max THI increases by one unit based on the historical minimum (64.8). The summer max THI has an inverse U-shaped effect with a threshold value of 72. Above that threshold, the increase in the summer max THI will decrease MPC. However, if the THI is below this threshold value, its increase will positively affect the MPC, because in that lower THI range, a warmer environment would benefit livestock production, as found in Du Preez et al. [64] and Correa-Calderon et al. [65]. Nationally, most regions experienced MPC reduction induced by heat stress, as the comparison between the historical summer max THI observations and the threshold shows more than 2/3 of past observations exceed the threshold value of 72.

The effects of other weather factors are also plotted in Figure 1. Fall precipitation also exhibits an inverse U-shaped relationship with MPC (Figure 1b), that is, MPC increases at a decreasing rate as fall precipitation rises up to the threshold value of 14 inches. Beyond that threshold value, MPC declines at an increasing rate. Generally, since about 90% of the fall precipitation observations are smaller than the threshold, to some extent, more precipitation would increase MPC, perhaps reflecting added forage availability [66,67]. For winter PDSI (Figure 1c), its total effect also shows an inverse U-shape with a threshold value of 0. This implies that as PDSI rises, indicating less drought, MPC tends to increase, but as PDSI continues to increase to be higher than the threshold value of 0, then MPC falls as the environmental condition gets wetter. These results agree with previous literature [68,69].

4.3. Model Comparison

Now we compare model performance by examining the in and out of sample model fit using the root mean squared error (RMSE) as the criterion [70]. The in-sample RMSE is computed with the data from 1950 to 2014, assessing how effective the models are in reproducing data, whereas the out-of-sample RMSE, calculated by the holdout dataset from 2015 to 2022, is to evaluate the forecasting accuracy of the models on a new dataset of independent variables [71]. The results in

Table 5. Model forecasting performance.

Model	In-Sample RMSE	Out-of-Sample RMSE
	(1950–2014)	(2015–2022)
Pooled OLS model	0.051	0.060
Fixed effects model	0.068	0.067
SEM	0.039	0.048
SDM	0.037	0.051

show that the estimates from the spatial models outperform those from the conventional panel data models.

Next, we statistically test differentiating parameters using a Wald test. First, we test whether the SEM spatial dependence (λ) factor is significantly different from zero and find it is at the 1% confidence level. In turn, we conclude the SEM model is more appropriate than the fixed effects panel model and that it is important to incorporate spatial correlation.

Next, we test whether $\phi +\gamma \tau =0$, which if so, means that SDM reduces to SEM [72]. This is also rejected at the 1% level, which indicates that SDM performs better than SEM. Thus, it is important to incorporate the correlation between the independent and the omitted variables.

5. Discussion

Our estimation results show changes in weather conditions influence U.S. milk production (on a per cow basis). Summer heat stress, as measured by summer max THI, exerts a significant effect on milk production. It is estimated to have an inverse U-shaped relationship with a threshold value of 72. When holding other variables constant, below this threshold, milk yield increases as the summer max THI rises from extreme cold weather conditions to the comfort thermal zoom of dairy cows, but when this THI continues increasing to be higher than that threshold value, milk productivity declines as dairy cows would experience the stress of heat and exhibit degraded production performance, which are aligned with findings by Renaudeau et al. [58]. Fall precipitation and winter drought conditions also exert an inverse U-shaped effect with threshold values of 14 inches and 0 on the PDSI scale, respectively. In essence, regarding such an inverse U-shaped relationship, an extreme weather condition, such as moving toward too much or too little rain, will negatively affect milk production.

On the methodological side, we use several model forms in an effort to reduce omitted variable bias and improve model fit. The results strongly support spatial panel models, particularly the SDM, that can control spatial correlation that arises due to omitted weather variables, which also provide promising evidence for future research to explore other potential spatial models, such as the spatial lag model (SLM) [73]. Our study considers the minimum and maximum values of different weather indicators, and the results can be considered lower and upper bounds for weather impacts on milk productivity, while further efforts should also be made to investigate the overall effects using measures such as the mean and median. It should be noted that the spatial panel models are designed to reduce the effect of omitted spatially pervasive forces but do not eliminate that bias. Therefore, in subsequent research, it would be good to explore improved model specifications and include more weather-related variables. Future work should also consider alternative model validation methods such as cross-validation and bootstrap methods [74,75], along with exploring the sensitivities of the choice of the time period for out-of-sample testing and the extent to which various combinations of explanatory variables would affect model results [76]. Additionally, there are different statistical tests that can be utilized to examine the spatial dependence [77,78] and future work should investigate the feasibility and suitability of different statistical tests to improve the reliability of modeling analysis.