Forecasting hotel demand uncertainty using time series Bayesian VAR models

Abstract

Demand uncertainty is a fundamental characteristic of the hospitality industry. Hotel room inventory is fixed, and devising an accurate daily demand measurement is a key operational challenge. In practice, it is difficult to predict the industry stability and capture demand uncertainty, so the industry relies on demand estimates. This process of estimation affects revenue maximization, as it is sensitive to incremental costs. In this article, we implemented vector autoregressive (VAR) models and compared them to the Bayesian VAR to examine the accuracy of predicting demand. We evaluated the results using a new measure of forecasting accuracy, the mean arctangent absolute percentage error (MAAPE). The results generated from the forecasts confirm the significant improvement in forecasting performance that can be obtained using the Bayesian model. It is noteworthy that the VAR performs the best for the lower horizons. The results also suggest that MAAPE outperforms other existing accuracy measures, in terms of error rates.

Keywords

Bayesian vector autoregressive models forecasting demand hotel occupancy forecast revenue management uncertainty

Introduction

Always dynamic and often responsive to the business environment, the hospitality industry tends to change whenever any kind of event creates uncertainty in the market. As such, this highly competitive industry requires different types of guiding principles for expected demand changes and their implications for future hotel prices, which can increase the variable costs, in financial terms. Forecasts of hotel statistics like occupancy, average daily rate (ADR), and revenue per available room are important when assessing market uncertainty. This type of demand estimation is sensitive to incremental costs that affect revenue maximization. Therefore, reducing forecast errors is a systematic process dependent on a number of external factors and, if executed correctly, can generate an increase in revenue. In a study for the airline industry, Pölt (1998) estimated that a 20% reduction of forecast error can be translated in a 1% incremental revenue increase.

Demand forecasts assess the impact of demand uncertainty when evaluating macro-predictability and micro-uncertainty factors. In today’s unpredictable environment, the question is how accurate forecasting predictions can be anywhere beyond a month in advance. Hoteliers also struggle with how to predict consumer behavior and, practically, restrict daily volatility due to unconstrained demand. With this in mind, the criteria for choosing forecasting methods and accuracy measures are challenged by the market, the length of forecast horizon, and the type of accuracy measures (Oh and Morzuch, 2005). New demand models need to be implemented (van Ryzin, 2005) to provide statistically sound and accurate forecast performance measures (Song and Li, 2008).

The seasonality of the hospitality industry also creates significant sales volatility. This volatility can cause uncertainty about whether a company has the capacity to maximize revenue and profit, and it also produces cash management challenges and the need for efficient operational planning in balancing supply and demand as appropriate to the season. Seasonal sales and demand fluctuate between off, middle, and peak seasons. Moreover, demand differs based on transient, or group segment, consumers and their lengths of stay. Revenue managers who develop forecasting estimations must consider any events scheduled both a short or long time in advance (e.g. conferences, meetings, games, concerts, etc.), in addition to the weekly sales fluctuations that are characteristic particularly of business or airport hotels. These types of properties receive a higher proportion of consumers from Tuesday to Thursday, while the occupancy falls dramatically from Friday to Monday. The fixed costs, however, remain the same.

In this framework, effective revenue management necessitates a certain elasticity in responding to market demand and differentiating room prices based on changes therein, in an attempt to maximize revenue through consumers’ willingness to pay. For this reason, revenue management systems utilize a variety of statistical methods to forecast demand based on historical data or other variables, such as price, discounts, weather, or economic indicators. In everyday hotel operations, the revenue managers are requested to forecast other metrics as well, such as the ADR, overbooking, cancellations, and no-show rooms, and, in some cases, the lengths of stays. When making forecasts, a manager can predict each arrival separately (a practice called disaggregate forecasting) and develop independent forecasts, or they can consider the total number of arrivals for a day (aggregate forecasting) and then develop forecasts by categorizing arrivals in a number of metrics (e.g. rate class, length of stay, historical data, etc.). A study by Weatherford et al. (2001) suggests that disaggregate forecasting is more accurate than aggregated forecasts. To this end, Thompson (2009) designed a tool to support hoteliers by identifying the best method of aggregation to use in their revenue management forecasts for room demand.

This article contributes to the literature by employing vector autoregressive (VAR) model and vector error correction model (VECM) using the Bayesian technique (Bayesian vector autoregressive (BVAR)) to estimate a probability distribution of forecast demand uncertainty. Despite the popularity of forecasting using the VAR approach, limited research has been conducted in the hospitality and tourism using the BVAR models specifically (see Gunter and Önder, 2015; Song et al., 2013; Wong et al., 2006). In addition, we evaluated the results using a new measure of forecasting accuracy, the mean arctangent absolute percentage error (MAAPE) as proposed by Kim and Kim (2016). This is different from the widely used mean absolute percentage error (MAPE) measure, and, to the best of our knowledge, no other study has used MAAPE in the hospitality and tourism literature. The study uses monthly hotel occupancy statistics, in addition to ADR statistics, which influence demand volatility, in London as a source of historical information. We use prior literature combined with historical data to propose an improved method of estimating forecast demand.

This study ultimately shows how to improve hotel occupancy forecasting. Our opening section discusses the relevant literature and then transitions to an overview of the different forecasting models and the forecasting accuracy measures. We then explain the estimation model. The article proceeds with an analysis of the study results, highlighting the employed techniques. Finally, we summarize the article’s conclusions and possible future research.

Background research

Forecasting is a central process in revenue management, is based on demand probabilities, and is conducted mainly to estimate aggregate demand and to optimize distribution (Talluri and van Ryzin, 2004). In the context of the hotel industry, forecasting using certain parameters is the basis by which revenue managers estimate cancellation rates, capacity allocation, no-show rates, overbooking, and pricing decisions. Forecasting demand is also widely used in the airline industry to estimate different booking patterns (constrained and unconstrained demands), as well as cancellation and no-show probabilities. Because of the common characteristics between the airline and the hotel industry (e.g. fixed capacity, consumer heterogeneity, perishable inventory, advance selling, and uncertain and fluctuating demand), in addition to similar trends in consumer behavior, demand forecasting may be employed in the latter industry as well. The most frequently used forecasting methods rely on historical data to make predictions. However, this may be a drawback in industries where the products change often or past data are of little use (mainly due to market economic crashes and industrial innovation), such as the airline, retail, and media industries.

In addition to the airline industry, there have been many studies of fields outside the hospitality and tourism industry in disciplines such as meteorology, statistics, actuarial, and finance that discuss forecasting performance and the importance of the proposed accuracy measures (see Armstrong and Collopy, 1992; Fildes and Makridakis, 1995; Hyndman and Koehler, 2006; Makridakis et al., 1982; Makridakis et al., 1998). These studies provide various forecasting accuracy measures, and the work by Hyndman and Koehler (2006) demonstrates an in-depth analysis of such measures. Demand fluctuations unique in the hospitality and tourism industry have a significant effect on accurate forecasting measurements; however, in response, a large number of researchers in both the tourism (see Crouch, 1994; Li et al., 2005; Song et al., 2010; Witt and Witt, 1992) and the hospitality literature (see Kimes, 1999; Law, 2004; Schwartz and Cohen, 2004; Weatherford et al., 2001; Zakhary et al., 2011) have worked to develop a fitting forecasting identity.

In general, in the hospitality and tourism forecasting literature, an antiquated set of accuracy measures is used, resulting often in contradictory conclusions. Unlike previous studies, this study primarily relies on MAAPE, a new accuracy measure, to assess the forecasting models. However, to evaluate its accuracy, we compare MAAPE to several other metrics that represent the common industry practice. Within the literature, scholars have focused on the methods (outlined in the following) that represent different ways to evaluate the accuracy of forecasting models’ parameters. For example, Song and Li (2008) discussed tourism forecasting demands using the MAPE and root mean square percentage error. Yüksel (2007) used MAPE, mean absolute deviation (MAD), and mean squared deviation. Rajopadhye et al. (2001) used the MAD and the MAPE to measure the performance of the forecast algorithm. Schwartz (1999) used MAD, MAPE, mean squared error (MSE), and standard deviation error to monitor the accuracy of hotels’ occupancy forecasts. Armstrong and Collopy (1992), after evaluating metrics for 11 models, recommended the use of geometric mean of the relative absolute error for a set of time series model parameters, the median Relative Absolute Error (RAE) when fewer series are available, and the median absolute percentage error otherwise. In addition, the aforementioned authors recommended that, for comparing accuracy across series, the root mean square error (RMSE) is not reliable and is therefore inappropriate. Koupriouchina et al. (2014) assessed a set of 17 forecasting accuracy measures, observing practical challenges and inconsistencies across the measures.

The process of evaluating the performance of a particular forecast model is based on performing a repeated task using specific set of indicators, I_t, selected by the forecaster (e.g. the history of occupancy or ADR) through time t to produce a forecast $f_{t} + H$ of expected occupancy $y_{t} + H$ that is observed at the future time $t + H$ . We assume that the forecast period H is 1 day and that t counts to daily timescale (Taylor, 2007). Hence, demand forecasts are based on the predictors that lead to a degree of error.

Forecasts based on econometrics models are subject to all types of uncertainty pertaining to the validity of their results, such as the appropriateness of the estimation model, the employed parameters, and the measures or policies considered. According to Taylor (2007), the accuracy of forecasts is evaluated by a certain loss function, $L (y_{t + 1}, f_{t + 1})$ . It defines the cost that occurs when we predict $f_{t + 1}$ , but the outcome is $y_{t + 1}$ . This difference is called the forecasting error, $e_{t + 1} = y_{t + 1} - f_{t + 1}$ , and is realized over time, in which the error is denoted as e_t at time t, y_t is the actual observations at time t, and f_t denote the forecast of y_t based on all previous observations at time t. The forecasting error is just as important as the forecast itself (Talluri and van Ryzin, 2004). In this study, the estimation and forecasting process was a rolling process. We used a data set at time t to obtain the parameters, and the estimation period is extended by 1 month $f_{t + 1}$ . The parameters were then be reestimated per period in the functional form of a forecast.

Forecasting hotel demand using VAR models

In order to forecast demand uncertainty, this study employed the VAR model, an econometrics instrument used for multivariate time series analyses. Since the 1980s, which saw the release of a seminal article by Litterman (1979) and a model extended by Sims (1989), VAR has been well-established as a primary means to develop macroeconomic forecasting models. The VAR model is a framework for describing the dynamic behaviors of the interrelationships between stationary variables (Pfaff, 2008), wherein each variable is a linear function of past lags of itself, as well as the past lags of the other variables involved. VAR has been used successfully in econometric forecasting and also in the tourism industry to forecast demand in various countries and tourism-generated employment in Denmark (see Song and Witt, 2000; Song and Witt, 2006; Song et al., 2003; Witt et al., 2004; Wong et al., 2006). Forecasts based on VAR models are flexible because they represent the conditional mean of a stochastic process of specified variables in the model (Zivot, 2006).

We used evidence found in previous studies to inform certain choices in the preparation stages for our study. For example, the study by Song and Witt (2006) used VAR to forecast tourism demand to Macau. Their study results suggest that the main market for Macau comes from mainland China, and the business sector should enhance its investments to fulfill the needs of tourists from China. Another study by Wong et al. (2006) examined the forecasting performance of Hong Kong tourism by employing various VAR derivative models. Specifically, the authors focused on the BVAR model to evaluate the demand, rather than the unrestricted VAR model. The study results indicated that the BVAR model produced better accuracy results when examining the forecasting performance than the unrestricted VAR models. There are some notable exceptions to this, such as the study by Witt et al. (2004) who employed various models to forecast the impact of tourism expenditure on employment in Denmark. The findings indicated that instead, the VECM generates the most accurate forecasting results.

A different study by Haensel and Koole (2011) aimed to forecast the accumulated booking curve and the number of expected reservations in a hotel using the Additive Holt–Winters (AHW) method and compared it against the VAR model. The study findings indicated that the mean values of the VAR outperform the AHW forecast. They used hotel reservation data from three regions provided by Bookit, an online travel agency (OTA). This kind of data set consists only of the available rooms offered to the OTA by each hotel, meaning it does not comprise the entire hotel supply or demand (i.e. if the hotel offers only 3, 5, and 10 rooms per day to the OTA, then the demand is based on those specific allocated rooms). In our article, we conducted our forecasting performance by analyzing the average historical hotel data of nine hotels located in the city center of London.

VAR model

A basic multivariate VAR model for a time series consists of a set of K endogenous variables $y_{t} = (y_{1 t}, \dots, y_{k t}, \dots, y_{K t})$ for $k = 1, \dots, K$ . The VAR(p)-process is then defined as

y_{t} = A_{1} y_{t - 1} + \dots + A_{p} y_{t - p} + u_{t} for k = 1, \dots, K

with A_i are (K × K) coefficient matrices to be predicted for $i = 1, \dots, p$ is the lag length of the VAR, and $u_{t}$ is the K vector of endogenous variables in dimensional process with E( $u_{t}$ ) = 0 and time invariant positive definite covariance matrix $E (u_{t} u_{t}^{T}) = Σ_{u}$ (Hill and Griffiths, 2011; Pfaff, 2008).

The advantages of using the VAR model are its relative simplicity and its forecasting capabilities. Testing for Granger noncausality tests and determining whether one variable is useful in forecasting another are very easy. In addition, all variables are endogenous, so there is no need to specify if the variables are endogenous or exogenous. However, VAR has been criticized by researchers because of its scarcity in economic theory. Moreover, there are numerous parameters associated with it. If there are K equations, one for each K variables and p lags of each of the variables in each equation, the number of $(K + p K^{2})$ parameters needs to be estimated. Finally, VAR does not offer cointegration, meaning that the series is nonstationary and exists in a linear combination that is stationary, meaning that estimates such as the mean, variance, and autocovariance are constant at various lags between the periods (Hyndmand and Athanasopoulos, 2014).

Vector error correction model

As mentioned earlier, the VAR model does not offer cointegration, meaning that the time series are nonstationary. The VAR model can be improved to incorporate consistent estimation among the series. In this case, the solution is to modify the system of equations using the VECM, as this model offers cointegrated variables. Therefore, the second type of forecasting model in this study is the VECM. As a result, a set of I(1) variables is cointegrated (r > 0; where r is the number of linearly independent cointegrating vectors), if a linear combination that is I(0) exists. If y_t is a vector of I(1) variables, we need to estimate the number of cointegrating relationships between the variables (Kilian and Lütkepohl, 2017). With that being the case, we may rewrite the VAR as a VECM, so subtracting $y_{t - 1}$ on both sides of the equation and rearranging terms yields the VECM

Δ y_{t} = Π y_{t - 1} + Γ_{1} Δ y_{t - 1} + \dots + Γ_{p - 1} Δ y_{t - p + 1} + u_{t}

where

Π = - (I_{K} - A_{1} - \dots - A_{p})

and

Γ_{i} = - (A i + 1 + \dots + A_{p}), for i = 1, \dots, p - 1

The left-hand side of the equation, I(0), requires that the right-hand side must $Π y_{t - 1}$ to be I(0). If the rank of $Π$ is r > 0, it may be expressed as $Π = α β^{'}$ , where α and β are two K × r matrices of rank r. Hence, the rank of $Π$ is called the cointegrating rank of the process y_t (Kilian and Lütkepohl, 2017). Substituting the matrix coefficients $α β^{'}$ for $Π$ in the equation yields to the VECM equation because it includes the lagged error correction term $α β^{'} y_{t - 1}$

Δ y_{t} = α β^{'} y_{t - 1} + Γ_{1} Δ y_{t - 1} + \dots + Γ_{p - 1} Δ y_{t - p + 1} + u_{t}

In this study, the VAR order is chosen by a model selection criteria process. Therefore, to choose the optimal number of lags p, we considered four different goodness-of-fit criteria such as those of the Akaike information criterion (AIC) (Akaike, 1981), Hannan–Quinn (HQ) (Hannan and Quinn, 1979), Schwarz (SC) (Schwarz, 1978), and the final prediction error (FPE) criterion (Lütkepohl, 2005). We tested the AIC for orders $p = 0, \dots,8$ on all data sets, as it measures the discrepancy between the given model and the true model. The AIC criterion is more accurate with monthly data, similar to this study’s data set (Ivanov and Kilian, 2001). In addition, we consider the SC and HQ to have a theoretical advantage over AIC because they provide consistent estimates of the true lag order, p (Lütkepohl, 2005). The results given in Table 1 identify two lags.

Table 1.

Unit root tests results.

Variables	Lag order $\hat{ρ}$	ADF	PP
Occupancy	1	−4.474***	−4.018***
ADR	2	−3.523**	−3.223**
Demand	2	−4.676***	−3.695**
Revenue	2	−4.308***	−3.329**

Note: ADF: augmented Dickey–Fuller; PP: Phillips–Perron tests; ADR: average daily rate. Results are statistically significant at *p < 0.05, **p < 0.01, and ***p < 0.001.

Benchmarks and naïve models

The random walk forecast, also referred to as the naïve model, is a fundamental benchmark for forecasting that is flexible and requires no parameter identification. This basic model should be outperformed by more complex methods to warrant the additional complexity (Hyndman and Athanasopoulos, 2014). Given the most recent actual observation y_t, the forecast for h steps ahead is calculated as

{\hat{y}}_{t + h} = y_{t}

These values work reasonably well in forecasting a number of macroeconomic and financial variables and should be used as a benchmark or as a starting point for further investigations (Canova, 2011).

Forecasters use a number of benchmarks to conduct empirical evaluations and produce base forecasts. Theil’s U-statistic, also called the “no-change forecast,” is a common benchmark that can be interpreted as the RMSE of the forecasting model divided by the RMSE of a no-change model (naïve, U = 1) (Makridakis et al., 1983). The Theil’s U-statistic is defined as

U = \sqrt{\frac{\sum_{t = 1}^{n - 1} {(\frac{F_{t + 1} - A_{t + 1}}{A_{t}})}^{2}}{\sum_{t = 1}^{n - 1} {(\frac{A_{t + 1} - A_{t}}{A_{t}})}^{2}}}

where U is Theil’s U-statistic, A _t is the actual outcome value at time t, and F _t is the forecast value at time t. If U = 1, the naïve model is as good as the forecasting model. If U > 1, the naïve method would produce better results than the proposed forecasting model, while U < 1 indicates that the forecasting model is better than the naïve method.

Forecasting using BVAR

BVAR forecasting models have been effective at generating accurate forecasts when forecasting economic variables at several levels, so they have frequently appeared in the macroeconomic forecasting literature (see Carriero et al., 2015; Clark, 2011; Sarantis and Stewart, 1995). However, limited research has been conducted in the hospitality and tourism studies using these types of models (see Gunter and Önder, 2015; Song et al., 2013; Wong et al., 2006).

The VAR models flexibility and ability to fit the data under a minimal set of conditions bring with it a risk of overfitting the data. Bayesian methods can address these problems by incorporating priors into a VAR model in order to reduce the number of parameters to be estimated, while improving the significant uncertainty about the future paths projected by the model (Canova, 2011; Doan et al., 1984). The priors refer to a set of data properties of the Minnesota prior (prior beliefs), introduced by Litterman (1979), which is centered on the assumption that each variable follows an independent random walk process, possibly with drift

y_{t} = c + y_{t - 1} + ε_{t}

which represents the macroeconomic behavior of an economic variable, thus reducing parameterization uncertainty and improving forecast accuracy (Karlsson, 2013).

The BVAR model is based on an unrestricted VAR model that is then modified as it is subjected to stochastic (Bayesian) restrictions (Spencer, 1993). Hence, BVAR represents a solution to improve the forecasting performance of unrestricted VARs into the issue of dimensionality by first controlling the issues associated with most VAR models. The Bayesian part of the model provides an additional set of restrictions through prior probability distribution functions. Specifically, the VAR(p)-process is defined as

y_{t} = A_{1} y_{t - 1} + \dots + A_{p} y_{t - p} + u_{t} for k = 1, \dots, K

with A_i are (K × K) coefficient matrices to be predicted for $i = 1, \dots, p$ where p is the lag length of the VAR and $u_{t}$ is the K vector of endogenous variables in dimensional process with E( $u_{t}$ ) = 0 and time invariant positive definite covariance matrix $E (u_{t} u_{t}^{T}) = Σ_{u}$ (Hill and Griffiths, 2011; Pfaff, 2008). Therefore, using an appropriate parameterization of the variables, it is possible to make the estimation reliable.

A Bayesian estimator can be achieved by assessing the posterior distribution over the parameters of a structural equation model (Lütkepohl, 2017). The first step in preparing the posterior distribution is the specification of a prior distribution over the parameters. Sims and Zha (1998) have developed an approach for imposing priors directly on the structural parameters, if they are of main interest and prior beliefs are available. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood. In this study, the hyperparameters of the models are set as follows: The overall tightness is set to 0.1 as recommended for small VAR systems, the relative cross-variable tightness to 0.5, reflecting and the lag decay parameter to 1, similar to Spencer (1993), Sims and Zha (1998), and Wong et al. (2006).

Measures of forecast errors

The model selection within forecasting is important in itself. The forecast’s success depends on correctly predicting errors while estimating an accurate future; and the aim is to maximize revenue by setting the best available rate for the forecasted demand. However, there is no single clear measurement method to predict the forecasting error (Song and Li, 2008), mainly due to the different criteria, characteristics, and demands of different forecasting models, while also assessing the quality of the forecast. In this study, an analysis of certain methods has been applied to measure the forecast accuracy.

Unlike previous studies, this study implemented a new forecast accuracy measure introduced in a research by Kim and Kim (2016), the MAAPE. According to Kim and Kim, MAAPE does not have a symmetric version, so it overcomes the division by zero concern and the undefined values using bounded influences for outliers and considering the ratio as an angle instead of a slope. The MAAPE process is defined as follows

MAAPE = \frac{1}{N} \sum_{t = 1}^{N} ({AAPE}_{t}) for t = 1, \dots, N

where the arctangent absolute percentage error (AAPE) for a given observation is defined as

{AAPE}_{t} = arctan (| \frac{A_{t} - F_{t}}{A_{t}} |)

where arctan is the arctangent (or inverse tangent) function, the A _t is the actual outcome value at period t, and F _t is the forecast value at period t.

Unlike the regular absolute percentage error, the arctangent absolute error approaches $\frac{π}{2}$ when division by zero occurs. The AAPE is undefined when $y_{t} = f_{t} = 0$ , which often can be found in an intermittent demand time series.

In addition to the MAAPE, we measured the forecasting uncertainty using other suggested methods. One important criterion preferred when evaluating the forecasting performance of a predictor or an estimator is the MSE (Carbone and Armstrong, 1982; Makridakis et al., 1998), as the optimal forecast is the conditional mean (Taylor, 2007). It calculates the MSE between the forecast and the expected outcomes. The MSE is defined as

MSE = \frac{1}{N} \sum_{t = 1}^{N} {(A_{t} - F_{t})}^{2}

where A _t is the actual outcome value at time t, and F _t is the forecast value at time t.

The predominant measure of forecasting accuracy, and the easiest to interpret, is the MAPE (Byrne, 2012). It is used as a primary accuracy measure in the Makridakis forecasting competition (M-competition) (Makridakis et al., 1982) and is widely accepted as a measure for forecasting room (Schwartz and Hiemstra, 1997; Weatherford and Kimes, 2003; Yüksel, 2007) and tourism demand (Witt and Witt, 1992). In this case, the MAPE is defined as

MAPE = \frac{1}{N} \sum_{t = 1}^{N} (| \frac{A_{t} - F_{t}}{A_{t}} | \times 100)

where N is the number of data points, and A _t and F _t represent the actual and forecast values at period t, $t = 1, 2, \dots, n$ , respectively. However, MAPE accuracy is limited when the actual observation of A _t is zero [0] or close to zero [0], at which point the MAPE is not defined, as it is numerically unstable and produces infinite or undefined values (Hyndman and Koehler, 2006). In this context, due to missing actual daily data, MAPE has some drawbacks. A number of scholars have attempted to find an alternative way to improve the forecasting accuracy.

Another measure of accuracy is the RMSE, which is the square root of the MSE. RMSE compares errors of different calculation models for the same forecast data set and the eventual outcomes. It is a scale-dependent measure, which is also considered to be a main drawback, as this produces low reliability (Hyndman and Koehler, 2006). The RMSE is defined as

RMSE = \sqrt{\sum_{t = 1}^{N} \frac{{(A_{t} - F_{t})}^{2}}{N}}

where A _t is the actual outcome value at time t and F _t is the forecast value at time t.

The choice of which accuracy measure is most appropriate to use correlates to the nature of the forecast (Talluri and van Ryzin, 2004; Witt and Witt, 1991). Each recommended method demonstrates drawbacks (e.g. not symmetric and problematic in calculation, as with MAPE; scale-independent, as with MSE; scale-dependent, as with RMSE); the proposed MAAPE avoids the scaling issues, but still not symmetric and is undefined when $y = f = 0$ . Because of that, it is reasonable to evaluate a forecast performance using more than one measure, mainly based on the relevant data scale.

To the best of our knowledge, this is the first study to use VAR models and evaluate the results by employing MAAPE. Therefore, we chose to compare the VAR model with an error correction mechanism, the VECM, and the BVAR model.

Unit root tests

Statistical forecasting methods are based on the assumption that the process of generating the time series is stationary. A time series is stationary if its properties are not affected by a change in the time of origin (i.e. if y_t is a stationary time series, then for all s, the distribution of $(y_{t}, \dots, y_{t} + s)$ does not depend on t) (Montgomery et al., 2015). This indicates that the mean, variance, and covariance of a stationary process do not depend upon time. Before working on the estimation of the time series models, one must choose from a variety of tests and examine the stationarity of the variables. Unit root tests can be used to determine the data stationarity; if the series is nonstationary, a data transformation is necessary. In this study, to test the stationarity of the series data, we first applied the augmented Dickey–Fuller (ADF) test (Dickey and Fuller, 1979), a coefficient test for detecting the presence of a unit root. The ADF test fits a model of the form

Δ y_{t} = β_{1} + β_{2} t + δ y_{t - 1} + \sum_{j = 1}^{p} γ_{j} Δ y_{t - j} + ε_{t}

where y_t is any time series variable, the $y_{t - 1}$ is the one period lag value of y_t, $Δ y_{t} = y_{t} - Y_{t - 1}$ , and t is the trend variable. The symbols β₁, β₂, δ, and γ are the parameters, and ε_t is a standard error of the test regression. The test statistic for $H_{0} : δ = 0$ (nonstationary) against the alternative $H_{1} : δ < 0$ , which is equivalent to $ρ < 1$ (stationary) (stata.com, 2018a).

Second, in addition to the ADF test, we conducted the nonparametric statistical method Phillips–Perron (PP) test (Phillips and Perron, 1988). This test assumes that a variable has a unit root under the null hypothesis, and the alternative is that the variable was generated by a stationary process (stata.com, 2018b). The PP test involves fitting the regression from the ADF test equation as

Δ y_{t} = α + β t + δ y_{t - 1} + ε_{t}

where $Δ y_{t}$ is the change in the number of a time series variable at time t, we may exclude the constant α or include a t time trend term, and the ε_t is a standard error term. As with the ADF test, the unit root hypothesis test statistics are $H_{0} : δ = 0$ against the alternative $H_{1} : δ < 0$ .

Empirical evaluation

Forecasting specification and data set

Using the average historical hotel data of nine hotels (Census Rooms: 3722) within the upscale and upper upscale categories, all located in the city center of London (specifically the Kensington-Chelsea area), United Kingdom, we conducted an empirical study to forecast future hotel room demand. The properties consisted of central hotels catering to different market segments (e.g. leisure, business, groups, government travelers, etc.). The data set was provided by Smith Travel Research, Inc. (STR) and included monthly average room nights sold (occupancy) for the period between January 2012 and January 2018. However, we ended the estimation in December 2016 and divided the data set into two segments: the ex post (i.e. historical data that explain observation already obtained) and the ex ante (i.e. variable values yet to be realized), as suggested by Pindyck and Rubinfeld (1998). We used the data from January 2017 to January 2018 to evaluate the model’s forecasting performance.

To fit the VAR models, we first decided on the variables to be included. The choice of the variables is influenced by the objective of the model, which, in this case, was to forecast the hotel occupancy and trace the uncertainty process. Hence, the model must include variables that, in day-to-day hotel operations, can be expected to have an impact on the hotel occupancy. This study therefore was planned to incorporate the following variables: the monthly average room nights sold (occupancy), n_t; the ADR, y_t; the available rooms (supply), m_t; the average demand, d_t; and the monthly average exchange rate, e_t. The exchange rate has a direct and indirect impact on the ADR as well as on the expected demand. We retrieved the monthly average exchange rate data between the British pound and the US dollar from the Bank of England for the study period as well.

We used the nonparametric Cox–Stuart test to identify the overall occupancy trend of the values obtained. The estimation shows a decrease of about 12% in monthly hotel occupancy. Figure 1 illustrates the initial time series (top), the estimated trend component (second from top), the estimated seasonal component (third from top), and the estimated irregular component (bottom), for the period from January 2012 to January 2018. We conducted the initial forecasting VAR models analysis using STATA and performed the BVAR analysis using R packages.

Figure 1.

Estimated trend, seasonal, and irregular components of the time series.

However, before implementing forecasting models, it is important to identify and test the estimators for stationarity. As an initial check, we performed two unit root tests, the ADF (Dickey and Fuller, 1979) and the PP (Phillips and Perron, 1988) tests, which could indicate whether the monthly data series are stationary. Table 1 shows the results of these tests. In both cases, we considered all variables of order one as (I(1)). The results yielded significant p-values, which rejects the presence of a unit root save for one variable; here, we found we could not reject the null hypothesis that the data are nonstationary. Therefore, as all variables were integrated, we were able to conclude that VAR analysis could be performed.

In addition, we ran lag-order selection diagnostics for both models to decide on the lag length. The lag-order process ran for a sample covering the period between January 2012 and December 2016, and we ultimately ascertained that the best model was the one with the smallest lag length. In the underlying VAR (p), the AIC, HQ, and FPE indicated two lags, whereas the SC criterion showed one lag. The results shown in Table 2 provide the respective lag lengths.

Table 2.

Lag-order selection criteria.

Lag	LL	LR	df	p	FPE	AIC	HQIC	SBIC
0	−2102.85				3.40E + 26	75.2803	75.3505	75.4612
1	−1543.37	1119	25	0.000	1.80E + 18	56.1919	56.6126	57.2769*
2	−1501.61	83.534	25	0.000	9.8e + 17*	55.5931*	56.3643*	57.5823
3	−1481.92	39.37*	25	0.034	1.30E + 18	55.7829	56.9047	58.6763
4	−1467.79	28.268	25	0.296	2.10E + 18	56.171	57.6433	59.9685

Note: FPE: final prediction error; AIC: Akaike information criterion; LL: Log Likelihood; LR: Likelihood Ratio; HQIC: Hannan-Quinn Information Criterion; SB: Schwarz Information Criterion.

After choosing the appropriate lag length, the next step was to test for cointegration relationships. We used the Johansen maximum likelihood test (Johansen, 1995) to determine the number of cointegrating relationships. Table 3 indicates the possible value of r, the number of cointegrating equations. It shows r = 2, meaning that two variables in this model are stationary. From this table, because the trace statistic at r = 0 of 132.4243 exceeds the critical value (5%) of 68.52, we could reject the null hypothesis of no cointegrating equations. Similarly, because the trace statistic at r = 1 exceeds its critical value (5%) of 47.21, we were able to reject the null hypothesis. On the other hand, with the trace statistic at r = 2 of 29.1770 as less than its critical value of 29.68, we could not reject the null hypothesis. Therefore, because Johansen’s method is to accept as $\hat{r}$ as the first r at which the null hypothesis could not be rejected, we accepted r = 2 as the number of cointegrating equations between these four variables. The “*” by the trace statistic at r = 2 indicates that this value of r fit with Johansen’s multiple-trace test procedure. Thus, we conclude that there are two cointegrating vectors.

Table 3.

Johansen tests for cointegration.

Maximum rank	Parms	LL	Eigenvalue	Trace statistic	5% critical value	1% critical value
0	30	−1636.3535		132.4243	68.52	76.07
1	39	−1607.588	0.62913	74.8933	47.21	54.46
2	46	−1584.7299	0.54534	29.1770*	29.68	35.65
3	51	−1574.5786	0.29534	8.8744	15.41	20.04
4	54	−1570.3281	0.13633	0.3734	3.76	6.65
5	55	−1570.1414	0.00642

Note: LL: Log Likelihood.

*denotes rejection of the hypothesis at the 0.05 level.

Further, applying the Granger causality test determined the causal relationships between occupancy and the other variables. In this framework, if a variable $y_{2 t}$ affects a variable $y_{1 t}$ , for example, it indicates that the former and latter variables of $y_{2 t}$ help improve the forecasting of $y_{1 t}$ (Lütkepohl, 2011). Table 4 provides the Granger causality test results. It shows strong evidence that lagged ADR, demand, and revenue together predict occupancy fluctuations (with p-values at 0.061, 0.0079, and 0.0724, respectively). In practice, this is expected, as the demand influences the growth of ADR, which in turn affects the hotel occupancy.

Table 4.

Granger causality test results.

Dependent variable: Occupancy
Excluded	χ ²	df	Probability > F
ADR	2.9698	2	0.0610
Demand	5.3754	2	0.0079
Revenue	2.7785	2	0.0724
Date	5.2E + 05	1	0.0000
ALL	81,425	7	0.0000

Note: ADR: average daily rate.

Granger causality depends mainly on the data set in question, so the interactions between the variables may not be causal. From a business standpoint, in day-to-day hotel operations, a revenue manager has to make decisions based on demand variance and operational risk, a constant challenge that affects the hotel performance and its profitability. Therefore, to examine the response to an impulse innovation or shock from one variable on the future value of another variable, impulse response analysis can be used to examine the changes. To identify the effects of a market shock on hotel occupancy, we sought to predict whether any correlation might exist between the occupancy growth and the study variables. In this study, we analyzed the dynamic effect of a shock between occupancy and other variables. It is expected that a relative increase in ADR would show a positive effect on the occupancy and vice versa. In Figure 2, the impulse response graph indicates the effect of the shock over an 18-month period. We examined how occupancy would respond to changes in ADR, to changes in demand, to results of its own shock, and to changes in revenue. The results indicate that a shock to ADR would lead to a permanent decline in the hotel occupancy, while, similarly, a sharp rise in demand results in an increase in the hotel occupancy, followed by a decrease and stabilization over time. In addition, an unexpected decline in hotel revenue (as depicted in the lower right graph) might produce a decline in the hotel occupancy, although it shows a marginal positive increase overtime, as well as the decline of the ADR, which, in the end, would lead to an even sharper decline in the total hotel occupancy. Finally, the lower left graph shows the change of a shock in occupancy on occupancy. It implies that a positive shock to occupancy causes an increase in occupancy, followed by a sharp decrease at the fifth month, peaking at a marginal increase, and so on, until the effect dies out after roughly 5 months. Figure 2 shows some interesting outcomes that suggest that, in every potential scenario, the impact of a shock is associated with a highly persistent change within the first 5 months, but a stabilization over time.

Figure 2.

Impulse response analysis.

Once the data set parameters were set and the diagnostic tests applied to the VAR models, we then created the forecast performance model of hotel occupancy. We ran a dynamic h-step-ahead forecast, moving the estimation one step ahead for each variable. The one-step-ahead forecast continuously reestimates the model and produces two-step-ahead forecasts, and the process continues for h periods. A set of h = 1 to 12-steps-ahead monthly forecasts were generated. This common form of forecasting is called dynamic forecasting because each step uses the estimations of the previous steps. With this, we assessed the performance of a number of models by the previously mentioned accuracy measures over the different forecast horizons. These forecasting models include the naïve random walk model, the VAR model, VECM, and the Bayesian model (BVAR). The Bayesian model was introduced in the tourism demand forecasting by Wong et al. (2006) with the main advantage that its estimation is not restricted by the integration order of the variables involved. The resultant Bayesian models were ranked against the other forecasting performance models.

Results

The objective of this study was to evaluate forecast performance for hotel occupancy using a new measure of forecasting accuracy (MAAPE). To evaluate the forecasting performance, we set the forecasting horizon i to 18 months. In hotel operations, a forecasting horizon of 18 months is considered expansive enough to estimate hotel occupancy; it also provides a sufficient enough estimation that seasonality is taken into account. The day-to-day hotel operations are affected by any recession, expansion, or other movement that can occur in the economic or business daily cycles. In practice, we assess on a case-by-case basis.

Table 5 presents descriptive statistics for the study variables (obs = 60). The average occupancy percentage for the hotels in the sample was 86.17%, with a maximum of 95.73%. The ADR was US$134.40, with a minimum of US$103.55 and a maximum of US$172.31. Taking into consideration that this study covered the critical period prior to and following the Brexit announcement (June 2016), both occupancy and ADR are considered very high.

Table 5.

Descriptive statistics of variables.

Variable	Period	Obs	Mean	Standard deviation	Min	Max
Occupancy	2012m1–2017m12	60	86.17	7.59	64.1	95.73
ADR	2012m1–2017m12	60	134.4	16.17	103.55	172.31
Demand	2012m1–2017m12	60	97,576.37	9264.79	73,958.00	109,496.00
Revenue	2012m1–2017m12	60	13,200,000	2,447,878	8,217,120	18,500,000

Note: ADR: average daily rate.

Figure 3(a) shows the dynamic forecast estimates, along with the widths of the confidence intervals for the periods. The graph illustrates that the confidence bandwidths on the VAR model are very large; in other words, the VAR models do not forecast very confidently. In the VECM, the confidence band results predict better confidence. The bandwidth at the beginning is small and close to the observed values, but over the long term, it changes and holds. We can observe that the predicted relative occupancy of the VAR model after the first period is considerably lower than the actual occupancy. Similarly, the VECM’s predicted occupancy rate was lower than that observed during this out-of-sample period, but generated lower frequency.

Figure 3.

Estimated hotel occupancy forecasts.

The same conclusion becomes evident in Figure 3(b), which illustrates a combined graph of the forecasting models and the actual hotel occupancy. We can observe that there are only marginal differences between the two forecasting models, providing some evidence that the models perform similarly. However, when comparing the actual occupancy against the combined forecasts, the differences are substantially greater. Both forecasting models were unable to predict the sharp increase in occupancy in March–April 2017 and failed to foresee the drop that occurred in July 2017, although the VAR model appears to predict the latter drop. In general, it shows that the VECM slightly outperforms the VAR model over time. Several studies have found that the VECM outperforms other forecasting models (e.g. Witt et al., 2004). However, we need to investigate whether any events in those periods could account for the unexpected results.

The same analysis is valid for the forecasting BVAR model as well, given that the variables exhibit a similar behavior. The resulting forecasts are plotted in Figure 4. The forecast expectations from the BVAR model are plausible and more closely match the uncertainty observed over the life of the series thus far.

Figure 4.

BVAR forecast model. BVAR: Bayesian vector autoregressive.

Table 6 presents the results of the forecasting models for the 18 months from January 2017 to June 2018. All forecasting models produced quite similar estimated occupancy except the BVAR model. The random walk model estimated a growth of 0.18%; the VAR model predicted marginal growth of 0.36%; the VECM predicted growth of 0.43%; and the BVAR model indicated only a marginal growth of 0.02% over the 18-month period.

Table 6.

Estimated hotel occupancy forecast models.

Forecast horizon	RW	VAR	VECM	BVAR
2017m1	84.24	81.69	82.51	86.53
2017m2	84.40	83.69	84.90	87.13
2017m3	84.55	85.21	86.09	88.54
2017m4	84.70	86.96	87.21	86.30
2017m5	84.86	87.91	87.88	88.10
2017m6	85.01	88.02	87.91	88.44
2017m7	85.17	87.70	87.77	89.29
2017m8	85.32	87.12	87.66	86.71
2017m9	85.48	86.64	87.63	88.02
2017m10	85.63	86.40	87.71	86.00
2017m11	85.78	86.39	87.85	87.83
2017m12	85.94	86.54	88.01	86.89
2018m1	86.09	86.71	88.16	85.75
2018m2	86.25	86.85	88.29	85.96
2018m3	86.40	86.91	88.41	88.00
2018m4	86.55	86.90	88.51	88.25
2018m5	86.71	86.85	88.62	86.94
2018m6	86.86	86.80	88.73	87.27

Note: RW: random walk; VAR: vector autoregressive; VECM: vector error correction model; BVAR: Bayesian VAR.

Comparison of forecasting performance

To examine the accuracy of the forecasting methods, we compared the results using five methods of measuring accuracy that have been previously recommended for use: MAAPE, MAPE, MSE, RMSE, and Theil’s U-statistic. We separated the historical data into two subsamples to use with the estimation model: An in-sample set of 60 monthly observations (i.e. January 2012 to December 2016) and the remaining 12 observations (i.e. January 2017 to December 2017) framed as an out-of-sample forecast, in which we evaluated the monthly forecasts. Following this, the specified models were used to generate forecasts 6 months ahead. For the forecasting models, a lag length of 2 was chosen based on in-sample tests. Both the Engle–Granger and Johansen approaches were used to estimate the cointegrating vectors. The forecast performance presents how and when both forecasting methods succeeded and failed. To assess the statistical significance of improvements in forecasting accuracy of each forecast (versus improvements due to sampling variability), we applied the test of predictive accuracy proposed by Diebold and Mariano (1995).

Table 7 summarizes the forecasts of hotel occupancy generated by the four different forecast models, that is, the classical VAR, the VECM, the BVAR, and the naïve model. For the two sets of forecast values, each column refers to a specific accuracy measure and the model with the highest accuracy is highlighted in italics. The forecast error rate is averaged across each method for the forecasting horizons. From the monthly forecast results in Table 7, we can observe that the differences in forecast accuracy within the models are marginal. However, the gap between the first and the last horizon and the remaining forecast horizons (i horizons) generated substantial differences.

Table 7.

Forecast error measures test results.

Forecast	MAAPE	Rank	MAPE	Rank	MSE	Rank	RMSE	Rank	Theil’s U-statistic	Rank
Forecast horizon (h = 1)
Naïve	11.294	3	12.051	3	82.091	3	9.060	3	0.045	1
VAR	8.641	1	8.636	1	42.412	1	6.495	1	0.731	3
VEC	9.721	2	9.725	2	53.749	2	7.314	2	0.823	4
BVAR	14.051	4	15.071	4	128.450	4	11.333	4	0.355	2
Forecast horizon (h = 2)
Naïve	0.236	2	6.277	3	41.135	3	6.414	3	0.170	1
VAR	0.171	1	4.504	1	21.247	1	4.598	1	0.490	3
VEC	0.548	3	5.398	2	27.303	2	5.210	2	0.557	4
BVAR	1.829	4	9.399	4	69.120	4	8.314	4	0.417	2
Forecast horizon (h = 3)
Naïve	0.644	4	4.803	3	28.215	3	5.312	3	0.527	3
VAR	0.397	2	3.384	1	14.512	1	3.797	1	0.489	2
VEC	0.057	1	3.643	2	18.209	2	4.255	2	0.549	4
BVAR	0.501	3	6.917	4	50.604	4	7.113	4	0.370	1
Forecast horizon (h = 6)
Naïve	1.096	4	5.621	4	31.191	4	5.585	4	1.094	4
VAR	0.509	2	3.384	1	11.892	1	3.456	1	0.591	1
VEC	0.531	3	3.493	2	13.609	2	3.695	2	0.634	2
BVAR	0.439	1	5.288	3	28.958	3	5.381	3	0.681	3
Forecast horizon (h = 12)
Naïve	0.595	1	7.062	4	46.293	3	6.804	3	1.215	4
VAR	0.653	2	3.814	1	16.517	1	4.069	1	0.745	1
VEC	0.792	4	3.816	2	17.755	2	4.221	2	0.776	2
BVAR	0.682	3	4.521	3	22.837	3	4.778	3	1.027	3

Note: MAAPE: mean arctangent absolute percentage error; MAPE: mean absolute percentage error; MSE: mean squared error; RMSE: root mean square error; VAR: vector autoregressive; VEC: vector error correction; BVAR: Bayesian VAR.

The results suggest the following conclusions. First, the forecasting performances appear to be nearly the same within the accuracy measures. For example, forecasts generated by the classical VAR model perform better compared with not only the VECM but also with the BVAR model, based on the accuracy measures for most of the forecasting horizons. This finding supports the studies by Witt et al. (2004) and Engle and Yoo (1987), which concluded that the VAR model generates more accurate forecasting results than other models. The study by Wong et al. (2006) also found that the simple time series models often outperform the advanced econometric models. On the other hand, it shows that controls for the variables’ parameters should be taken into consideration.

Second, we examined MAAPE against the more traditional measure for errors in forecasting, MAPE. As discussed, results show that MAAPE outperformed MAPE. Within the data of Table 7, we can compare the forecasting performances for each model at different forecasting horizons to indicate the model’s performance in the separate short-term horizons. These findings are particularly noteworthy because MAAPE is a new accuracy metric in testing and helps to overcome the problem of division by zero, since, conversely, the MAPE metric produces infinite or undefined values due to missing actual data when the denominator is zero or close to zero (A _t = 0).

In addition, the forecasting performance of the models measured by MAAPE outperformed all other accuracy measure specifications. We also compared the performance of MAAPE against the MSE and RMSE measures (see Table 7 for details). As previously, MAAPE outperformed for both sets of forecast values, although the Theil’s U-statistic measures are close to the MAAPE measures. Nevertheless, this result highlights the superiority of MAAPE’s accuracy in predicting hotel occupancy.

Finally, as indicated by the results in Table 7, Theil’s U-statistic shows that the VAR model outperforms the VECM. Otherwise, the BVAR model performs better on short-term horizons, similar to the naïve model. In addition, the VAR forecasting model outperforms the naïve model as far the forecast horizon increases.

Conclusion

Accurate forecasts of the fluctuations in hotel occupancy are important when assessing expected returns that may subsequently contribute to revenue maximization. A remarkable variety of forecasting methods exists, but many of the conclusions obtained from these methods tend to be inaccurate or contradictory, reflecting the fact that forecasts depend on parameters. Because of this, it is an ongoing challenge for revenue managers to assess the effectiveness of different forecasting methods, with ultimate accuracy being the main criterion. In this article, in order to forecast demand uncertainty, we relied on four forecasting models. We compared the forecasting performances using monthly hotel data for 6 years and employing a variety of accuracy measures, including MAAPE, MAPE, MSE, RMSE, and Theil’s U-statistic, values over time.

This article contributes to the literature by employing VAR model VECM and using the Bayesian technique (BVAR) to estimate a probability distribution of forecast demand uncertainty. To the best of our knowledge, limited research has been conducted in the hospitality and tourism literature using BVAR models. Instead, the literature so far has studied time series models that focus on smoothing methods, ARIMA, and Holt–Winters models. In addition, the majority of studies uses accuracy measures that are numerically unstable when the actual observation A _t is zero, or close to zero (e.g. MAPE). In this study, we employed a new measure, MAAPE, suggested by Kim and Kim (2016) as the best means to measure forecasting accuracy. To the best of our knowledge, no other study has used the MAAPE metric. Hence, this article makes a contribution to the literature by analyzing a number of different forecasting models based on MAAPE, and ultimately demonstrates that MAAPE outperforms the other accuracy measures, in terms of error rates.

The determination of which forecasting method outperforms the others is based on the implied model restrictions. Thus, the use of unrestricted VAR models was incorporated into a practical approach using actual hotel data. Given the results of Table 7, we gain a rather good understanding of the forecasting models over the different forecast horizons. The results for the naïve, vector error correction, and VAR models are mixed, because each forecast density is off-center over several horizons and can still be further improved. This confirms that would always be an outperformed model where there will be evidence that have better coverage on short-term or long-term forecasting performance based on the data set availability and the model complexity. In addition, the simple time series models often outperform the BVAR models, based on the accuracy measures for most of the forecasting horizons (Wong et al., 2006). Moreover, the model performance is associated with the evaluation period chosen.

However, this study suffers from certain limitations that need to be addressed. The first is that the size of the data set is relatively small, so a larger sample size would be better for a more robust analysis. Second, the validation sample should be larger in order to be usable as a validation sample for long-term forecasting horizons. Third, MAAPE is a new metric, still to be investigated in further studies using other forecasting models. Despite these limitations, the study findings are informative and can be utilized as a basis for further research. The importance of occupancy forecasting is obvious, and a combination of forecasting methods might provide more efficient and accurate results. Therefore, whether it is subject-specific or common across the hotels needs further identification.

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