Forecasting Weekly Shipments of Hass Avocados from Mexico to the United States Using Econometric and Vector Autoregression Models

Abstract

Domestic production cannot meet the U.S. demand for avocados, satisfying only 10% of the national demand. Due to year-round production and longer shelf-life, the Hass variety of avocados accounts for about 85% of avocados consumed in the United States and roughly 95% of total avocado imports, primarily from Mexico. Using weekly data over the period July 3, 2011, to October 24, 2021, econometric and vector autoregression models are estimated regarding the seven main shipment sizes of Hass avocados from Mexico to the United States. Both types of models discern the impacts of inflation-adjusted and exchange-rate adjusted prices per box as well as U.S. disposable income, holidays and events, and seasonality on the level of Hass avocado shipments by size. In general, these impacts are robust across the respective models by shipment size. These types of models also mimic the variability in the level of shipments by size quite well based on goodness-of-fit metrics. Based on absolute percent error, these models provide reasonably accurate forecasts of the level of Hass avocado shipments from Mexico by size associated with a time horizon of 13 weeks. But neither type of models provides better forecast performance universally across all avocado shipment sizes.

1. Introduction

“Self-styled “prophets” who mislead us should be reminded that among the ancient Scythians, when prophets predicted things that failed to come true, they were laid, shackled hand and foot, on a little cart filled with heather and drawn by oxen, on which they were burned to death”-Unknown. “In science and in real economic life, it is terribly important not to be wrong much” [1].

Avocado is the fruit of the avocado tree, scientifically known as Persea Americana. This fruit is sought after because of its high nutrient value and often is added to various dishes due to its appealing flavor and rich texture. Avocado is the main ingredient in guacamole. The avocado has become an incredibly popular food among health-conscious individuals, often referred to as a superfood [2]. Per capita consumption of fresh avocados has increased markedly from 2.21 pounds in 2000 to 9.05 pounds in 2020 [3]. This surge in per capita consumption in roughly 20 years is slightly more than 300%.

In the United States, three commercial avocado regions are evident: Southern California, Florida, and Hawaii. Among these three areas, California produces the majority of the avocados followed by Florida and Hawaii. However, domestic production cannot meet the U.S. demand for avocados, satisfying only 10% of the national demand for avocados [4]. Due to year-round production and longer shelf-life, the Hass variety of avocados is the dominant and the most popular commercial type. Hass avocados account for about 85% of avocados consumed in the United States and roughly 95% of total avocado imports, primarily from Mexico, the major producer of avocados in the world [5, 6]. As such, we concentrate solely on the demand for Hass avocados.

2. Objectives

The objectives of this investigation are twofold: (1) to develop econometric and vector autoregression (VAR) models associated with the seven main shipment sizes of Hass avocados from Mexico to the United States; and (2) to provide ex-post forecasts over a period of 13 weeks out-of-sample. The main purpose of this investigation is to determine which class of models yields the better forecasts of weekly shipments. This analysis is of utmost importance to the Mexican Hass Avocado Importer Association (MHAIA) as well as stakeholders in the avocado industry in general.

The historical data used to estimate the respective models span the period with the week ending July 3, 2011, to the week ending October 24, 2021, a total of 539 observations. Based on these model specifications, we derive ex-post weekly forecasts over the week ending October 31, 2021, to the week ending January 23, 2022. Because the forecasts were generated over a period for which we have actual historical data, we are in position to determine their accuracy. Metrics used to determine forecast accuracy typically include root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percent error (MAPE) [1, 7, 8, 9]. Because the levels of avocado shipments are not the same across the respective sizes, in this analysis, attention is centered exclusively on MAPE. With MAPE, forecast accuracy is devoid of units of measurement.

3. Methodology

The econometric models consider the direct effects of specific market variables on weekly shipment levels of Hass avocados to the United States by size. Seven sizes (32, 36, 40, 48, 60, 70, and 84) of Hass avocados historically have accounted for close to 99% of all shipments since July 2011. The respective sizes refer to the number of avocados per box. The seven econometric models are single-equation relationships which account for seasonality, changes in real U.S. disposable personal income, changes in the Mexican peso to U.S. dollar exchange rate, changes in the real price per box of avocados shipped, inertia of shipments (a one-period lag of shipments), and qualitative events such as Cinco de Mayo, the Super Bowl, holidays (July 4/Independence Day, Thanksgiving, and Christmas), beginning of the month, end of the month, end of the year, the pandemic, and work stoppages.

Mathematically, the econometric model specification for this analysis is as follows:

$${\mathbf{lnY}}_{\mathbf{it}}={\mathbf{\beta }}^{\prime }\mathbf{ln}{\mathbf{X}}_{\mathbf{it}}+{\mathbf{\alpha }}^{\prime }{\mathit{Z}}_{\mathbf{it}}+{\mathbf{\epsilon }}_{\mathbf{it}},\mathrm{i}=1,2,\dots ,\mathrm{N},\mathrm{t}=1,2,\dots ,\mathrm{T}$$E1

where lnY_it is the logarithmic transformation of shipments of Hass avocados from Mexico to the United States by size i at time t, lnX_it is a column vector of logarithmic transformations of the continuous explanatory variables for size i in time t. Z_it corresponds to additional explanatory variables, namely indicator variables which correspond to the previously mentioned qualitative events. ${\mathbf{\alpha }}^{\prime }$ and β${}^{\prime }$ are the conformable vectors of parameters to be estimated, and ε_it is a column vector of error terms. As a result of the use of logarithmic transformations, β${}^{\prime }$ also represents the elasticities associated with the continuous explanatory variables.

According to Sims [10], one may consider equation (1) as multiple economic time series where lags (to be determined from the data and a priori knowledge) of each variable are allowed to affect the current position of each series. The general statement of the vector autoregressive model (VAR) is given as:

$${\mathrm{x}}_{\mathrm{t}}=\sum _{\mathrm{k}=1}^{\mathrm{K}}\mathrm{\alpha }\left(\mathrm{k}\right){\mathrm{x}}_{\mathrm{t}-\mathrm{k}}+{\mathrm{e}}_t,$$E2

where α(k) is an autoregressive matrix of dimension (nxn) at lag k which connects x_t and x_t-k, the vector of endogenous variables and lagged endogenous variables, n represents the number of endogenous variables included in the model, and e_t is a vector residual term of dimension (nx1). Most of the autoregressive parameters α(k) are equal to zero and K is the maximum lag based on model selection criteria such as the Akaike, Schwarz, and Hannan-Quinn information criteria (AIC, SIC, and HQC). In this analysis, the endogenous variables included in the VAR are the logarithmic transformation of shipments of Hass avocados from Mexico to the United States by size at time t. Hence, like the econometric models, the VAR consists of seven equations. In the VAR, we also include as exogenous variables real U.S. disposable personal income, the Mexican peso to U.S. dollar exchange rate, and the real price per box of avocados shipped as well as the qualitative variables previously described. Hence this specification technically is a structural vector autoregression model (SVAR).

Unit root tests, based on the use of Augmented Dickey-Fuller (ADF) tests, were conducted prior to the estimation of the VAR. In all cases, the respective endogenous variables are stationary or I(0). Thus, the appropriate model is a VAR in levels specification. Because the respective endogenous variables are stationary in levels, the examination of co-integration is superfluous.

To determine whether estimated coefficients are significantly different from zero, we adopt a level of significance of 0.10 for the econometric models and for the VAR model. This choice of the level of significance is conservative in terms of determining the key factors associated with shipments of Hass avocados from Mexico, especially given the number of weekly observations in this analysis.

3.1 Historical Avocado Shipments by Size

Historical weekly avocado shipments, the dependent variables in this investigation, in metric tons by size over the period July 3, 2011, to October 24, 2021, are shown in Figure 1. Shipments have increased over this period in all size classes, although they fluctuate noticeably from week to week. As well, the weekly deviations can be quite severe. Additionally, the avocado shipments of each size show definite seasonal patterns. In general, shipments for most sizes are seasonally lowest in July when the Normal and Marzeña harvests wind down and when the Loca and Aventajada harvests begin. Shipments tend to hit a peak each year in January in preparation for the Super Bowl, historically the largest avocado consuming season in the United States.

Figure 1.

Weekly Shipments of Hass Avocados from Mexico to the United States by Size, July 3, 2011, to October 24, 2021. Source: Mexican Hass Importer Association [11] and the Hass Avocado Board [12].

The descriptive statistics associated the dependent variables associated with the respective class of models is exhibited in Table 1. The label Q_SIZE_ refers to the shipment size measured in metric tons. On average, the weekly shipments vary from 515.53 metric tons (size 32) to 5,631.28 metric tons (size 48). The average share sizes of the respective weekly shipments over the 539 weekly periods are 3.46 percent for size 32; 5.79 percent for size 36; 9.99 percent for size 40; 39.61 percent for size 48; 22.83 percent for size 60; 12.19 percent for size 70; and 6.12 percent for size 84. Consequently, the two main sizes of weekly Hass avocado shipments are 48 and 60, combining for slightly more than 60 percent of total avocado shipments from Mexico to the United States.

Descriptive statistic	Q_SIZE_32	Q_SIZE_36	Q_SIZE_40	Q_SIZE_48	Q_SIZE_60	Q_SIZE_70	Q_SIZE_84
Mean	515.53	827.13	1,432.76	5,631.28	3,258.39	1,772.76	900.99
Median	413.02	747.75	1 345.58	5,420.38	3,074.43	1,689.01	845.76
Maximum	2,239.92	2875.40	4 101.05	14,523.68	8,103.01	4,454.53	2,375.21
Minimum	9.23	58.27	118.68	723.03	329.01	128.68	50.26
Std. Dev	399.08	491.78	751.86	2,366.38	1,460.29	887.89	497.03

Table 1.

Descriptive Statistics of the Weekly Shipments of Hass Avocados from Mexico to the United States by Size in Metric Tons, July 3, 2011, to October 24, 2021. Source: Calculations by the author.

4. Empirical results

Due to space limitations, the estimated parameters, standard errors, and p-values of the econometric models and the vector autoregression model are not reported. This information however is available from the author upon request.

Each of the respective econometric models is estimated by ordinary least squares (OLS) using the software packages EVIEWS 11.0.¹ The VAR model is estimated by seemingly unrelated regression (SUR) using the software package EVIEWS 11.0. Based on model selection criteria, the optimal lag length chosen in the VAR model is 1.

Both classes of models fit the historical weekly shipments well based on their goodness-of-fit (R² and adjusted R²) statistics. The respective econometric models explain between 85% and 93% of the weekly variability of avocado shipments, while the respective VAR model explains between 86% and 94% of the weekly variability of avocado shipments. Given the variability inherent in weekly shipments, simply put, the econometric models and the VAR model replicate the behavior of historical shipments quite well.

The continuous explanatory variables include lags of the logarithmic transformations of the avocado shipments. The econometric models include only the lag of the dependent variable in a particular equation, but the VAR model includes lags of all dependent variables in all equations. Both sets of models include the logarithmic transformation of real (inflation-adjusted) disposable income in the United States multiplied by the Mexico peso to U.S. dollar exchange rate and the logarithmic transformation of inflation-adjusted prices per box (in U.S. dollars) multiplied by the Mexican peso to U.S. dollar exchange rate. Consequently, the coefficients associated with inflation- and exchange-rate-adjusted disposable personal income and prices per box are elasticities.

Indicator variables associated with each calendar month are included to account for seasonality. The base or reference category is the month of July. As well, indicator variables are included to account for holidays, work stoppages, the beginning and ending of each month, the end of the calendar year, and the pandemic. The qualitative variable associated with the pandemic is equal to 1 beginning March 8, 2020, through October 24, 2021, and 0 otherwise. Finally, influential data points (outliers and leverage points) based on R-student statistics and hat diagonal elements also are accounted for with the use of indicator variables [14].

Impulse responses provide the impact of a one-time change in the “impulse variable” on the “response” variable over the course of several periods. In this analysis, the impulse variable is a particular endogenous variable in the system that pertains to the magnitude of Hass avocado shipments from Mexico of a certain size; the response variable refers to the magnitudes of any of the other remaining Hass avocado shipments from Mexico of other sizes. The number of periods to consider for the impulse-response functions is arbitrary. In this analysis, 13 weeks (one quarter) are considered. The impulse response functions associated with the VAR analysis are exhibited in Figure 2.

Figure 2.

Impulse Response Functions Associated with the VAR Analysis.

The variance decomposition of a particular endogenous variable indicates the percentage of its forecast error variance explained by shocks attributed to other endogenous variables in the system. Again, the number of periods to consider is arbitrary. Like the situation for the impulse response functions, 13 weeks (one quarter) are considered. For any period, the percentages associated with the respective endogenous variables must sum to 1. The variance decompositions associated with the VAR analysis are exhibited in Figure 3.

Figure 3.

Variance Decompositions Associated with the VAR analysis.

Most of the forecast error variance (between 81 and 99 percent) associated with Hass avocados of size 32 is explained by itself. The forecast error variance associated with Hass avocados of size 36 is explained by the volume of Hass avocados of size 32 (between 67 and 73 percent) and size 36 (between 19 and 28 percent). The forecast error variance associated with Hass avocados of size 40 is attributed to the volume of Hass avocados of size 32 (between 60 and 69 percent), size 36 (between 13 and 15 percent), and size 40 (between 17 and 21 percent). The forecast error variance associated with Hass avocados of size 48 is explained by the volume of Hass avocados of size 32 (between 52 and 56 percent), size 36 (between 13 and 16 percent), and size 48 (between 20 and 22 percent). The forecast error variance associated with Hass avocados of size 60 is attributed to the volume of Hass avocados of size 32 (between 32 and 40 percent), size 48 (between 28 and 33 percent), and size 60 (between 14 and 18 percent). The forecast error variance associated with Hass avocados of size 70 is explained by the volume of Hass avocados of size 32 (between 22 and 32 percent), size 48 (between 20 and 24 percent), size 60 (between 12 and 26 percent), and size 70 (roughly 17 percent). Finally, the forecast error variance associated with Hass avocados of size 84 is attributed to the volume of Hass avocados of size 32 (between 15 and 23 percent), size 48 (between 14 and 21 percent), size 60 (between 13 and 27 percent), size 70 (between 9 and 17 percent), and size 84 (between 22 and 27 percent).

Seasonal troughs are evident in July for all sizes in both the econometric models and the VAR model0. Additionally, avocado shipments from Mexico are higher in January, March, April, September, October, November, and December relative to July for all sizes.

The effects of exchange rate and inflation are embedded in the U.S. dollar per box price. For the econometric models, the own-price elasticity (responsiveness) of the respective sizes are as follows: size 32, -0.2081; size 36, -0.1864; size 40, -0.1490; and size 48, -0.1156. The own-price elasticities monotonically decrease in size (in absolute value) with increases in box size. No price responsiveness is evident for shipments of sizes 60, 70 and 84. Consequently, the prices per box associated with these shipment sizes were dropped from those econometric models. The VAR model, unlike the econometric models, includes prices per box for all sizes of avocado shipments. But not all coefficients associated with the respective prices are significantly different from zero. Non-significant coefficients regarding prices per box were dropped in the VAR model. Only 13 of the 49 coefficients associated with prices are significantly different from zero in the VAR model. The prices of sizes 40, 48, 60, and 70 impact avocado shipments of size 32. The prices of sizes 40 and 48 affect avocado shipments of size 36. The price of size 32 impacts avocado shipments of size 40, while the prices of sizes 40 and 84 impact avocado shipments of size 48. Prices of the respective box sizes do not affect avocado shipments of size 60. The prices of sizes 60 and 70 impact avocado shipments of size 70, and the prices of sizes 60 and 84 impact avocado shipments of size 84. The statistically significant price elasticities in the VAR model range from -0.4169 to -0.4985. Bottom line, for both the econometric and VAR models, Hass avocado shipments from Mexico to the United States are not sensitive to inflation-adjusted and exchange-rate adjusted prices per box.

On the other hand, avocado shipments are responsive to changes in U.S disposable personal income, adjusted for inflation and exchange rates. The income elasticity of the respective sizes from the econometric models are as follows: size, 32 0.4162; size 36, 0.3489; size 40, 0.3000; size 48, 0.4172; size 60, 0.3736; size 70, 0.3772; and size 84, 0.2613. The income elasticities of the respective sizes from the VAR model are as follows: size, 32 0.3995; size 36, 0.3639; size 40, 0.3475; size 48, 0.4001; size 60, 0.3649; size 70, 04142; and size 84, 0.2733. Consequently, the estimates of the respective income elasticities are robust across the econometric and VAR models.

Further, in the econometric models, all estimated coefficients associated with the lagged dependent variables are not only positive but also between 0 and 1. This finding also is evident in the VAR model concerning the lag of the dependent variable in question. Hence, this result confirms inertia or persistence in weekly shipments by size of Hass avocados from Mexico to the United States across the econometric and VAR models.

That said, the VAR model, unlike the econometric models, includes one-period lags for all sizes of avocado shipments. But like the situation for prices per box, not all coefficients associated with the respective lags are significantly different from zero. Non-significant coefficients regarding lags of the dependent variables were dropped in the VAR model. Only 29 of the 49 coefficients associated with lags are significantly different from zero in the VAR model.

We consider holiday/calendar events associated with The Super Bowl, Cinco de Mayo, July 4 (Independence Day), Thanksgiving, and Christmas as potential determinants of avocado shipments from Mexico. Importantly, we recognize that lags occur with respect to these holiday/calendar events. As such, we do not consider contemporaneous impacts of the respective holiday/calendar events, but we allow the lags associated with these events to vary from one to four weeks. Subsequently, we choose the optimal lag based on the model selection criteria once again. For The Super Bowl, the optimal lag length is three weeks across all shipment sizes; for Cinco de Mayo, Thanksgiving, and Christmas, the optimal lag length is two weeks across all shipment sizes. The lag length for Independence Day varies from one to four weeks depending on the shipment size in the econometric models. Based on model selection criteria, the lag length for Independence Day is the same (two weeks) across all shipment sizes in the VAR model.

The Super Bowl and the Christmas holiday season have substantial impacts on avocado shipments of all sizes over the historical period of analysis. For the econometric models, the Super Bowl boosts avocado shipments during the three weeks leading up to that event by 26.70 percent (size 70) to 42.61 percent (size 32). For the VAR model, the Super Bowl boosts avocado shipments during the three weeks leading up to that event by 25.90 percent (size 70) to 40.92 percent (size 32). For the two weeks leading up to Christmas, avocado shipments increase from 32.49 percent (size 84) to 53.29 percent (size 36) based on the econometric models. For the two weeks leading up to Christmas, avocado shipments increase from 24.88 percent (size 84) to 48.96 percent (size 36) based on the VAR model.

The Cinco de Mayo, Thanksgiving, and Independence Day holidays deliver much smaller lifts to weekly avocado shipments. For the two weeks leading up to Cinco de Mayo, this lift varies from 6.55 percent (size 40) to 12.92 percent (size 32) in the econometric models and from 6.72 percent to 13.47 percent (size 32) in the VAR model; for the two weeks leading up to Thanksgiving, this lift ranges from 4.23 percent (size 32) to 8.99 percent (size 70) in the econometric models and from 1.90 percent (size 32) to 9.08 percent (size 70) in the VAR model; and for the weeks leading up to Independence Day, this lift varies from 2.71 percent (size 48) to 17.08 percent (size 84) in the econometric models and from 2.63 percent (size 48) to 13.84 percent (size 84) in the VAR model.

Based on the econometric models, work stoppages diminish avocado shipments from 20.95 percent (size 70) to 37.19 percent (size 32); based on the VAR model, work stoppages diminish avocado shipments from 23.24 percent (size 70) to 32.25 percent (size 32). Based on the econometric models, avocado shipments at the end of each calendar year are lower from 19.82 percent (size 32) to 23.73 percent (size 48) on average. Based on the VAR model, avocado shipments at the end of each calendar year are lower from 18.49 percent (size 32) to 24.20 percent (size 60) on average. At the beginning of each month avocado shipments are lower by 3.79 percent (size 84) to 8.94 percent (size 60) on average based on the econometric models. At the beginning of each month avocado shipments are lower by 4.98 percent (size 84) to 8.38 percent (size 60) on average based on the VAR model. At the end of each month avocado shipments are lower by 4.82 percent (size 84) to 8.33 percent (size 40) on average in the econometric models. At the end of each month avocado shipments are lower by 4.84 percent (size 84) to 8.60 percent (size 40) on average in the VAR model.

Finally, the pandemic affects only avocado shipments of sizes 32, 36, 40, and 48 based on the econometric models. For these respective sizes, avocado shipments are lower by 5.19 percent (size 48) to 8.36 percent (size 36). No statistically significant impacts are evident for avocado shipments of sizes 60, 70, and 84 concerning the pandemic based on the econometric models. The pandemic affects only avocado shipments of sizes 36, 40, 48, and 70 based on the VAR model. For these respective sizes, avocado shipments are lower by 3.68 percent (size 70) to 7.38 percent (size 36). No statistically significant impacts are evident for avocado shipments of sizes 32, 60, and 84 concerning the pandemic based on the VAR model. Thus, the set of models provides different impacts of the pandemic on weekly Hass avocado shipments.

4.1 Weekly Ex-Post Forecasts of Avocado Shipments

We derive ex-post forecasts of weekly avocado shipments by size using the estimated econometric models and the VAR model. That is, the weekly observations from July 3, 2011, to October 24, 2021, serve as the training sample. The weekly observations from October 31, 2021, to January 23, 2022, constitute the out-of-sample period during which all endogenous and predetermined variables are known. This 13-week period then serves as the ex-post forecast time horizon. By comparing the ex-post forecasts with the actual values of avocado shipments by size for this 13-week period, we are in position to measure forecast accuracy/performance based on absolute percent error.

4.1.1 Ex-Post Weekly Forecasts of the Econometric Models: October 31, 2021, to January 23, 2022

The weekly ex-post forecasts of avocado shipments by size for the econometric models in this analysis are exhibited in Table 2. The out-of-sample mean absolute percent error (MAPE) by size of avocado shipments over the 13-week period is as follows: (1) 14.16 percent for size 32; (2) 13.09 percent for size 36; (3) 8.65 percent for size 40; (4) 7.63 percent for size 48; (5) 9.09 percent for size 60; (6) 8.47 percent for size 70; and (7) 12.73 percent for size 84. If we sum the avocado shipments by size over the 13-week period, the absolute percent error (APE) is noticeably reduced to: (1) 2.18 percent for size 32; (2) 4.31 percent for size 36; (3) 2.09 percent for size 40; (4) 0.44 percent for size 48; (5) 2.23 percent for size 60; (6) 5.48 percent for size 70; and (7) 10.89 percent for size 84. In addition, we find that the econometric models over forecast shipment sizes of 32, 60, 70, and 84, and under forecast shipment sizes of 36, 40, and 48.

Week Ending	Size 32 Econometric Model Forecasts	Size 32 Econometric Model Actual Values	Size 32; APE; %	Size 36 Econometric Model Forecasts	Size 36 Econometric Model; Actual Values	Size 36 APE; %
10/31/2021	381.60	336.35	13.45	635.60	565.23	12.45
11/07/2021	336.49	382.97	12.14	565.03	761.05	25.76
11/14/2021	334.83	519.00	35.49	558.43	895.00	37.61
11/21/2021	336.49	368.29	8.63	557.15	677.06	17.71
11/28/2021	318.00	367.31	13.42	517.33	545.13	5.10
12/05/2021	329.31	350.39	6.02	550.34	690.14	20.26
12/12/2021	374.24	379.44	1.37	629.69	580.51	8.47
12/19/2021	402.75	349.18	15.34	657.73	632.71	3.95
12/26/2021	345.88	335.48	3.10	547.81	554.58	1.22
01/02/2022	422.79	363.72	16.24	669.44	636.30	5.21
01/09/2022	585.68	480.00	22.02	950.59	805.00	18.09
01/16/2022	574.66	491.75	16.86	902.52	796.81	13.27
01/23/2022	567.92	473.26	20.00	867.31	857.61	1.13
MAPE			14.16			13.09
Total Shipments Over the 13-Week Period	5,310.64	5,197.14	2.18	8,608.97	8,997.13	4.31
Share	1.83	1.83		2.97	3.17
	Size 40; Econometric Model Forecasts	Size 40 Econometric Model Actual Values	Size 40; APE; %	Size 48 Econometric Model Forecasts	Size 48 Econometric Model Actual Values	Size 48; APE; %
10/31/2021	1,450.10	1,318.65	9.97	8,768.08	7,201.39	21.76
11/07/2021	1,333.00	1,534.83	13.15	7,602.40	8,325.94	8.69
11/14/2021	1,351.25	1,983.00	31.86	7,546.83	10,059.00	24.97
11/21/2021	1,373.03	1,469.48	6.56	7,538.97	7,488.08	0.68
11/28/2021	1,281.08	1,350.85	5.16	6,976.40	6,844.98	1.92
12/05/2021	1,315.29	1,318.78	0.26	7,193.09	7,423.32	3.10
12/12/2021	1,460.27	1,501.56	2.75	8,066.19	7,541.12	6.96
12/19/2021	1,528.73	1,530.78	0.13	8,283.57	8,035.67	3.08
12/26/2021	1,239.62	1,353.04	8.38	6,463.23	6,749.94	4.25
01/02/2022	1,444.44	1,412.01	2.30	7,139.76	7,341.54	2.75
01/09/2022	2,057.41	1,789.00	15.00	9,962.37	8,936.00	11.49
01/16/2022	1,987.84	1,834.51	8.36	9,736.56	9,005.40	8.12
01/23/2022	1,937.06	1,784.96	8.52	9,601.86	9,471.32	1.38
MAPE			8.65			7.63
Total Shipments Over the 13-Week Period	19,759.12	20,181.45	2.09	104,879.31	104,423.70	0.44
Share	6.81	7.11		36.13	36.78

Week Ending	Size 60 Econometric Model Forecasts	Size 60 Econometric Model; Actual Values	APE; %	Size 70 Econometric Model Forecasts	Size 70 Econometric Model; Actual Values	Size 70; APE; %
10/31/2021	7,026.11	5,304.07	32.47	4,201.42	3,259.63	28.89
11/07/2021	6,025.71	6,226.88	3.23	3,602.64	3,311.36	8.80
11/14/2021	5,953.48	6,954.00	14.39	3,478.10	3,599.00	3.36
11/21/2021	5,917.22	5,264.35	12.40	3,395.57	2,951.70	15.04
11/28/2021	5,484.31	5,028.20	9.07	3,114.68	2,709.93	14.94
12/05/2021	5,635.94	5,304.49	6.25	3,228.04	3,008.71	7.29
12/12/2021	6,306.59	5,748.60	9.71	3,583.05	3,343.45	7.17
12/19/2021	6,389.35	6,620.06	3.49	3,558.31	3,465.78	2.67
12/26/2021	4,928.56	5,683.14	13.28	2,770.05	3,105.52	10.80
01/02/2022	5,198.11	5,528.54	5.98	2,841.29	2,942.64	3.44
01/09/2022	7,097.62	6,818.00	4.10	3,846.23	3,710.00	3.67
01/16/2022	7,137.48	6,886.07	3.65	3,939.41	3,918.44	0.54
01/23/2022	7,172.62	7,159.17	0.19	4,014.78	3,879.23	3.49
MAPE			9.09			8.47
Total Shipments Over the 13-Week Period	80,273.10	78,525.57	2.23	45,573.57	43,205.39	5.48
Share	27.65	27.66		15.70	15.22

	Size 84; Econometric Model Forecasts	Size 84; Econometric Model; Actual Values	Size 84; APE %
10/31/2021	2,291.82	1,793.39	27.79
11/07/2021	2,057.29	1,672.39	23.01
11/14/2021	2,020.47	1,899.00	6.40
11/21/2021	1,997.70	1,587.21	25.86
11/28/2021	1,858.10	1,537.38	20.86
12/05/2021	1,889.44	1,509.03	25.21
12/12/2021	2,036.63	1,939.23	5.02
12/19/2021	2,039.79	1,852.42	10.11
12/26/2021	1,581.65	1,667.00	5.12
01/02/2022	1,580.41	1,511.98	4.53
01/09/2022	2,120.85	1,962.00	8.10
01/16/2022	2,186.93	2,147.58	1.83
01/23/2022	2,241.58	2,279.70	1.67
MAPE			12.73
Total Shipments Over the 13-Week Period	25,902.66	23,358.31	10.89
Share	8.92	8.23

Table 2.

Summary of Ex-Post Forecasts from the Econometric Models of Weekly Shipments by Size, October 31, 2021, to January 23, 2022.

4.1.2 Ex-Post Weekly Forecasts of the VAR Model: October 31, 2021, to January 23, 2022

The weekly ex-post forecasts of avocado shipments by size for the VAR model in this analysis are exhibited in Table 3. The out-of-sample mean absolute percent error (MAPE) by size of avocado shipments over the 13-week period is as follows: (1) 17.06 percent for size 32; (2) 11.09 percent for size 36; (3) 10.13 percent for size 40; (4) 6.99 percent for size 48; (5) 8.36 percent for size 60; (6) 13.07 percent for size 70; and (7) 22.27 percent for size 84. If we sum the avocado shipments by size over the 13-week period, we find that the VAR model over forecasts avocado shipments of sizes 32, 60, 70, and 84, and under forecast avocado shipments of sizes 36, 40, and 48. The absolute percent error (APE) associated with the sum of the avocado shipments by size is as follows: (1) 11.80 percent for size 32; (2) 2.67 percent for size 36; (3) 9.70 percent for size 40; (4) 3.05 percent for size 48; (5) 0.99 percent for size 60; (6) 12.09 percent for size 70; and (7) 22.21 percent for size 84.

Week Ending	Size 32; VAR Model Forecasts	Size 32; VAR Model; Actual Values	Size 32 APE; %	Size 36; VAR Model Forecasts	Size 36; VAR Model; Actual Values	Size 36; APE; %
10/31/2021	376.22	336.35	11.85	606.61	565.23	7.32
11/07/2021	363.6	382.97	5.06	575.93	761.05	24.32
11/14/2021	384.92	519.00	25.83	593.17	895.00	33.72
11/21/2021	401.82	368.29	9.10	614.48	677.06	9.24
11/28/2021	384.97	367.31	4.81	577.44	545.13	5.93
12/05/2021	394.46	350.39	12.58	599.61	690.14	13.12
12/12/2021	434.22	379.44	14.44	661.05	580.51	13.87
12/19/2021	453.36	349.18	29.84	678.43	632.71	7.23
12/26/2021	384.5	335.48	14.61	558.19	554.58	0.65
01/02/2022	448.5	363.72	23.31	654.62	636.30	2.88
01/09/2022	612.35	480.00	27.57	930.26	805.00	15.56
01/16/2022	589.94	491.75	19.97	864.9	796.81	8.55
01/23/2022	581.37	473.26	22.84	842.2	857.61	1.80
MAPE			17.06			11.09
Total Shipments Over the 13-Week Period	5,810.23	5,197.14	11.80	8,756.89	8,997.13	2.67
Share	2.00	1.83		3.02	3.17
	Size 40; VAR Model Forecasts	Size 40; VAR Model; Actual Values	Size 40; APE; %	Size 48; VAR Model Forecasts	Size 48; VAR Model; Actual Values	Size 48; APE; %
10/31/2021	1,351.96	1,318.65	2.53	8,412.14	7,201.39	16.81
11/07/2021	1,279.05	1,534.83	16.67	7,412.38	8,325.94	10.97
11/14/2021	1,300.26	1,983.00	34.43	7,387.08	10,059.00	26.56
11/21/2021	1,324.29	1,469.48	9.88	7,415.18	7,488.08	0.97
11/28/2021	1,233.30	1,350.85	8.70	6,849.73	6,844.98	0.07
12/05/2021	1,244.27	1,318.78	5.65	7,067.63	7,423.32	4.79
12/12/2021	1,362.11	1,501.56	9.29	7,901.41	7,541.12	4.78
12/19/2021	1,418.74	1,530.78	7.32	8,108.15	8,035.67	0.90
12/26/2021	1,133.86	1,353.04	16.20	6,286.76	6,749.94	6.86
01/02/2022	1,298.45	1,412.01	8.04	6,915.74	7,341.54	5.80
01/09/2022	1,841.03	1,789.00	2.91	9,538.17	8,936.00	6.74
01/16/2022	1,741.06	1,834.51	5.09	8,986.52	9,005.40	0.21
01/23/2022	1,696.26	1,784.96	4.97	8,958.66	9,471.32	5.41
MAPE			10.13			6.99
Total Shipments Over the 13-Week Period	18,224.64	20,181.45	9.70	101,239.55	104,423.70	3.05
Share	6.28	7.11		34.87	36.78

	Size 60; VAR Model Forecasts	Size 60; VAR Model; Actual Values	Size 60; APE; %	Size 70; VAR Model Forecasts	Size 70; VAR Model; Actual Values	Size 70; APE; %
10/31/2021	6,640.06	5,304.07	25.19	3,859.67	3,259.63	18.41
11/07/2021	5,804.66	6,226.88	6.78	3,374.57	3,311.36	1.91
11/14/2021	5,804.97	6,954.00	16.52	3,406.20	3,599.00	5.36
11/21/2021	5,795.43	5,264.35	10.09	3,446.34	2,951.70	16.76
11/28/2021	5,431.96	5,028.20	8.03	3,274.24	2,709.93	20.82
12/05/2021	5,648.83	5,304.49	6.49	3,531.62	3,008.71	17.38
12/12/2021	6,343.34	5,748.60	10.35	4,008.05	3,343.45	19.88
12/19/2021	6,478.06	6,620.06	2.14	4,027.62	3,465.78	16.21
12/26/2021	4,967.52	5,683.14	12.59	3,138.62	3,105.52	1.07
01/02/2022	5,238.02	5,528.54	5.25	3,210.35	2,942.64	9.10
01/09/2022	7,084.24	6,818.00	3.90	4,298.21	3,710.00	15.85
01/16/2022	6,942.89	6,886.07	0.83	4,311.87	3,918.44	10.04
01/23/2022	7,120.34	7,159.17	0.54	4,542.62	3,879.23	17.10
MAPE			8.36			13.07
Total Shipments Over the 13-Week Period	79,300.32	78,525.57	0.99	48,429.98	43,205.39	12.09
Share	27.32	27.66		16.68	15.22

Week Ending	Size 84; VAR Model Forecasts	Size 84; VAR Model; Actual Values	Size 84; APE; %
	Size 84	Size 84	Size 84
10/31/2021	2,085.34	1,793.39	16.28
11/07/2021	1,923.59	1,672.39	15.02
11/14/2021	1,936.66	1,899.00	1.98
11/21/2021	1,946.07	1,587.21	22.61
11/28/2021	1,885.98	1,537.38	22.67
12/05/2021	2,074.63	1,509.03	37.48
12/12/2021	2,330.93	1,939.23	20.20
12/19/2021	2,403.42	1,852.42	29.74
12/26/2021	1,895.91	1,667.00	13.73
01/02/2022	1,910.83	1,511.98	26.38
01/09/2022	2,622.32	1,962.00	33.66
01/16/2022	2,680.18	2,147.58	24.80
01/23/2022	2,850.00	2,279.70	25.02
MAPE			22.27
Total Shipments Over the 13-Week Period	28,545.86	23,358.31	22.21
Share	9.83	8.23

Table 3.

Summary of Ex-Post Forecasts from the Vector Autoregression Models of Weekly Shipments by Size, October 31, 2021, to January 23, 2022.

Based on MAPE, the econometric models provide better out-of-sample forecasting accuracy of avocado shipment sizes of 32, 40, 70, 84. But the VAR model provides better out-of-sample forecast accuracy of avocado shipments of sizes 36, 48, and 60. If we sum avocado shipments by size over the 13-week ex-post period, based on APE, the econometric models provide better out-of-sample forecasting accuracy for sizes 32, 40, 48, 70 and 84. The VAR model yields better out-of-sample forecast performance for sizes 36 and 60 if we sum avocado shipments by size over the 13-week ex-post period. Hence, neither the econometric models nor the VAR model provides better forecast accuracy universally across all avocado shipment sizes. As well, as exhibited in Tables 2 and 3, the forecasted and actual shares of the respective shipment sizes align very well across the board over the 13-week period from October 31, 2021, to January 23, 2022.

5. Concluding remarks

Both the econometric models and the VAR model allow us to discern the impacts of inflation-adjusted and exchange-rate adjusted prices per box as well as inflation-adjusted and exchange-rate adjusted U.S. disposable income, holidays and events, and seasonality on the level of Hass avocado shipments by size. In general, these impacts are robust across the class of models by shipment size. As well, the respective class of models mimic the variability in the level of shipments by size quite well based on goodness-of-fit metrics. Moreover, based on absolute percent error, the respective class of models provide reasonably accurate forecasts of the level of Hass avocado shipments by size. Going forward, we recommend generating weekly ex-ante forecasts on a continual basis based on the econometric models and the VAR model. These respective forecasts would provide lower and upper bounds of the level of Hass avocado shipments from Mexico to the United States by size to stakeholders in the industry.

Acknowledgments

We gratefully acknowledge the Mexican Hass Avocado Importer Association who provided financial support as well as the data for this study.

Conflict of interest

No conflicts of interest exist concerning this study.

Jel classification

JEL Codes: C13, C32, C53.