Sales forecasting using multi-equation transfer function models

The purpose of this study is first, to demonstrate how multivariate forecasting models can be effectively used to generate high performance forecasts for typical business applications. Second, this study compares the forecasts generated by a simultaneous transfer function model (STF) model and a white noise regression model with that of a univariate ARIMA model. The accuracy of these forecasting models is judged using their residual variances and forecasting errors in a post-sample period. It is found that ignoring the residual serial correlation can greatly degrade the forecasting performance of a multi-variable model, and in some situations, cause a multi-variable model to perform inferior to a univariate ARIMA model. This paper also demonstrates how a forecaster can use an STF model to compute both the multi-step ahead forecasts and their variances easily.     

Dynamic relationship analysis of us gasoline and crude oil prices

This paper studies the dynamic relationships between US gasoline prices, crude oil prices, and the stock of gasoline. Using monthly data between January 1973 and December 1987, we find that the US gasoline price is mainly influenced by the price of crude oil. The stock of gasoline has little or no influence on the price of gasoline during the period before the second energy crisis, and seems to have some influence during the period after. We also find that the dynamic relationship between the prices of gasoline and crude oil changes over time, shifting from a longer lag response to a shorter lag response. Box-Jenkins ARIMA and transfer function models are employed in this study. These models are estimated using estimation procedure with and without outlier adjustment. For model estimation with outlier adjustment, an iterative procedure for the joint estimation of model parameters and outlier effects is employed. The forecasting performance of these models is carefully examined. For the purpose of illustration, we also analyze these time series using classical white-noise regression models. The results show the importance of using appropriate time-series methods in modeling and forecasting when the data are serially correlated. This paper also demonstrates the problems of time-series modeling when outliers are present.     

Forecasting growth with time series models

This paper compares the structure of three models for estimating future growth in a time series. It is shown that a regression model gives minimum weight to the last observed growth and maximum weight to the observed growth in the middle of the sample period. A first-order integrated ARIMA model, or 1(1) model, gives uniform weights to all observed growths. Finally, a second-order integrated ARIMA model gives maximum weights to the last observed growth and minimum weights to the observed growths at the beginning of the sample period. The forecasting performance of these models is compared using annual output growth rates for seven countries.     

Sales revenues: Time-series properties and predictions

This paper compares the predictive ability of ARIMA models in forecasting sales revenue. Comparisons were made at both industry and firm levels. With respect to the form of the ARIMA model, a parsimonious model of the form (0, 1, 1) (0, 1, 1) was identified most frequently for firms and industries. This model was identified previously by Griffin and Watts for the earnings series, and by Moriarty and Adams for the sales series. As a parsimonious model, its predictive accuracy was quite good. However, predictive accuracy was also found to be a function of the industry. Out of the eleven industry classifications, ‘metals’ had the lowest predictive accuracy using both firmspecific and industry-specific ARIMA models.     

Forecasting with an econometric model: The ‘ragged edge’ problem

In practical econometric forecasting exercises, incomplete data on current and immediate past values of endogenous variables are available. This paper considers various approaches to this ‘ragged edge’ problem, including the common device of treating as ‘temporarily exogenous’ an endogenous variable whose value is known, by deleting it from the set of endogenous variables for whose forecast values the model is solved and suppressing the corresponding structural equation. It is seen that this forecast can be adjusted to coincide with the optimal forecast. The initial discussion concerns the textbook linear simultaneous equation model; extensions to non-linear dynamic models are described.     

On the forecasting of high-frequency financial time series based on ARIMA model improved by deep learning

Through empirical research, it is found that the traditional autoregressive integrated moving average (ARIMA) model has a large deviation for the forecasting of high-frequency financial time series. With the improvement in storage capacity and computing power of high-frequency financial time series, this paper combines the traditional ARIMA model with the deep learning model to forecast high-frequency financial time series. It not only preserves the theoretical basis of the traditional model and characterizes the linear relationship, but also can characterize the nonlinear relationship of the error term according to the deep learning model. The empirical study of Monte Carlo numerical simulation and CSI 300 index in China show that, compared with ARIMA, support vector machine (SVM), long short-term memory (LSTM) and ARIMA-SVM models, the improved ARIMA model based on LSTM not only improves the forecasting accuracy of the single ARIMA model in both fitting and forecasting, but also reduces the computational complexity of only a single deep learning model. The improved ARIMA model based on deep learning not only enriches the models for the forecasting of time series, but also provides effective tools for high-frequency strategy design to reduce the investment risks of stock index.     

Can cointegration‐based forecasting outperform univariate models? An application to Asian exchange rates

Conventional wisdom holds that restrictions on low‐frequency dynamics among cointegrated variables should provide more accurate short‐ to medium‐term forecasts than univariate techniques that contain no such information; even though, on standard accuracy measures, the information may not improve long‐term forecasting. But inconclusive empirical evidence is complicated by confusion about an appropriate accuracy criterion and the role of integration and cointegration in forecasting accuracy. We evaluate the short‐ and medium‐term forecasting accuracy of univariate Box–Jenkins type ARIMA techniques that imply only integration against multivariate cointegration models that contain both integration and cointegration for a system of five cointegrated Asian exchange rate time series. We use a rolling‐window technique to make multiple out of sample forecasts from one to forty steps ahead. Relative forecasting accuracy for individual exchange rates appears to be sensitive to the behaviour of the exchange rate series and the forecast horizon length. Over short horizons, ARIMA model forecasts are more accurate for series with moving‐average terms of order >1. ECMs perform better over medium‐term time horizons for series with no moving average terms. The results suggest a need to distinguish between ‘sequential’ and ‘synchronous’ forecasting ability in such comparisons. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimal forecast combination based on ensemble empirical mode decomposition for agricultural commodity futures prices

Improving the prediction accuracy of agricultural product futures prices is important for investors, agricultural producers, and policymakers. This is to evade risks and enable government departments to formulate appropriate agricultural regulations and policies. This study employs the ensemble empirical mode decomposition (EEMD) technique to decompose six different categories of agricultural futures prices. Subsequently, three models—support vector machine (SVM), neural network (NN), and autoregressive integrated moving average (ARIMA)—are used to predict the decomposition components. The final hybrid model is then constructed by comparing the prediction performance of the decomposition components. The predicting performance of the combination model is then compared with the benchmark individual models: SVM, NN, and ARIMA. Our main interest in this study is on short-term forecasting, and thus we only consider 1-day and 3-day forecast horizons. The results indicate that the prediction performance of the EEMD combined model is better than that of individual models, especially for the 3-day forecasting horizon. The study also concluded that the machine learning methods outperform the statistical methods in forecasting high-frequency volatile components. However, there is no obvious difference between individual models in predicting low-frequency components.     

Forecasting with latent structure time series models: an application to nominal interest rates

In this paper we develop a latent structure extension of a commonly used structural time series model and use the model as a basis for forecasting. Each unobserved regime has its own unique slope and variances to describe the process generating the data, and at any given time period the model predicts a priori which regime best characterizes the data. This is accomplished by using a multinomial logit model in which the primary explanatory variable is a measure of how consistent each regime has been with recent observations. The model is especially well suited to forecasting series which are subject to frequent and/or major shocks. An application to nominal interest rates shows that the behaviour of the three‐month US Treasury bill rate is adequately explained by three regimes. The forecasting accuracy is superior to that produced by a traditional single‐regime model and a standard ARIMA model with a conditionally heteroscedastic error. Copyright © 1999 John Wiley & Sons, Ltd.     

Neural network versus econometric models in forecasting inflation

Artificial neural network modelling has recently attracted much attention as a new technique for estimation and forecasting in economics and finance. The chief advantages of this new approach are that such models can usually find a solution for very complex problems, and that they are free from the assumption of linearity that is often adopted to make the traditional methods tractable. In this paper we compare the performance of Back‐Propagation Artificial Neural Network (BPN) models with the traditional econometric approaches to forecasting the inflation rate. Of the traditional econometric models we use a structural reduced‐form model, an ARIMA model, a vector autoregressive model, and a Bayesian vector autoregression model. We compare each econometric model with a hybrid BPN model which uses the same set of variables. Dynamic forecasts are compared for three different horizons: one, three and twelve months ahead. Root mean squared errors and mean absolute errors are used to compare quality of forecasts. The results show the hybrid BPN models are able to forecast as well as all the traditional econometric methods, and to outperform them in some cases. Copyright © 2000 John Wiley & Sons, Ltd.     

A unified view of statistical forecasting procedures

A large number of statistical forecasting procedures for univariate time series have been proposed in the literature. These range from simple methods, such as the exponentially weighted moving average, to more complex procedures such as Box–Jenkins ARIMA modelling and Harrison–Stevens Bayesian forecasting. This paper sets out to show the relationship between these various procedures by adopting a framework in which a time series model is viewed in terms of trend, seasonal and irregular components. The framework is then extended to cover models with explanatory variables. From the technical point of view the Kalman filter plays an important role in allowing an integrated treatment of these topics.     

Prior Information and ARIMA Forecasting

Using the method of ARIMA forecasting with benchmarks developed in this paper, it is possible to obtain forecasts which take into account the historical information of a series, captured by an ARIMA model (Box and Jenkins, 1970), as well as partial prior information about the forecasts. Prior information takes the form of benchmarks. These originate from the advice of experts, from forecasts of an annual econometric model or simply from pessimistic, realistic or optimistic scenarios contemplated by the analyst of the current economic situation. The benchmarks may represent annual levels to be achieved, neighbourhoods to be reached for a given time period, movements to be displayed or more generally any linear criteria to be satisfied by the forecasted values. The forecaster may then exercise his current economic evaluation and judgement to the fullest extent in deriving forecasts, since the laboriousness experienced without a systematic method is avoided.     

Co-integration,error correction and improved medium-term regional VAR forecasting

This study investigates possible improvements in medium-term VAR forecasting of state retail sales and personal income when the two series are co-integrated and represent an error-correction system. For each of North Carolina and New York, three regional vector autoregression (VAR) models are specified; an unrestricted two-equation model consisting of the two state variables, a five-equation unrestricted model with three national variables added and a Bayesian (BVAR) version of the second model. For each state, the co-integration and error-correction relationship of the two state variables is verified and an error-correction version of each model specified. Twelve successive ex ante five-year forecasts are then generated for each of the state models. The results show that including an error-correction mechanism when statistically significant improves medium-term forecasting accuracy in every case.     

Forecasting time series with increasing seasonal variation

Four options for modeling and forecasting time series data containing increasing seasonal variation are discussed, including data transformations, double seasonal difference models and two kinds of transfer function-type ARIMA models employing seasonal dummy variables. An explanation is given for the typical ARIMA model identification analysis failing to identify double seasonal difference models for this kind of data. A logical process of selecting one option for a particular case is outlined, focusing on issues of linear versus non-linear increasing seasonal variation, and the level of stochastic versus deterministic behavior in a time series. Example models for the various options are presented for six time series, with point forecast and interval forecast comparisons. Interval forecasts from data-transformation models are found to generally be too wide and sometimes illogical in the dependence of their width on the point forecast level. Suspicion that maximum likelihood estimation of ARIMA models leads to excessive indications of unit roots in seasonal moving-average operators is reported.     

Factor forecasts for the UK

Data are now readily available for a very large number of macroeconomic variables that are potentially useful when forecasting. We argue that recent developments in the theory of dynamic factor models enable such large data sets to be summarized by relatively few estimated factors, which can then be used to improve forecast accuracy. In this paper we construct a large macroeconomic data set for the UK, with about 80 variables, model it using a dynamic factor model, and compare the resulting forecasts with those from a set of standard time‐series models. We find that just six factors are sufficient to explain 50% of the variability of all the variables in the data set. These factors, which can be shown to be related to key variables in the economy, and their use leads to considerable improvements upon standard time‐series benchmarks in terms of forecasting performance. Copyright © 2005 John Wiley & Sons, Ltd.     

Volatility forecasting for stock market incorporating macroeconomic variables based on GARCH-MIDAS and deep learning models

Empirical experiments have shown that macroeconomic variables can affect the volatility of stock market. However, the frequencies of macroeconomic variables are low and different from the stock market volatility, and few literature considers the low-frequency macroeconomic variables as input indicators for deep learning models. In this paper, we forecast the stock market volatility incorporating low-frequency macroeconomic variables based on a hybrid model integrating the deep learning method with generalized autoregressive conditional heteroskedasticity and mixed data sampling (GARCH-MIDAS) model to process the mixing frequency data. This paper firstly takes macroeconomic variables as exogenous variables then uses the GARCH-MIDAS model to deal with the problem of different frequencies between the macroeconomic variables and stock market volatility and to forecast the short-term volatility and finally takes the predicted short-term volatility as the input indicator into machine learning and deep learning models to forecast the realized volatility of stock market. It is found that adding macroeconomic variables can significantly improve the forecasting ability in the comparison of the forecasting effects of the same model before and after adding the macroeconomic variables. Additionally, in the comparison of the forecasting effects among different models, it is also found that the forecasting effect of the deep learning model is the best, the machine learning model is worse, and the traditional econometric model is the worst.     

Forecasting the European Credit Cycle Using Macroeconomic Variables

We question the ability of macroeconomic data to predict risk appetite and ‘flight‐to‐quality’ periods in the European credit market using a model inspired by the Markov switching literature. This model allows for a direct mapping of exogenous variables into state probabilities. We find that various surveys and transformed hard data have a forecasting power. We show that despite its depth, the 2008–2009 crisis should not be regarded as an unusual episode that would have to be modelled by an additional state. Finally, we show that our model outperforms a pure Markov switching model in terms of forecasting accuracy, thus clearly indicating that economic figures are helpful in forecasting the credit cycle. Copyright © 2011 John Wiley & Sons, Ltd.     

Forecasting Aggregates with Disaggregate Variables: Does Boosting Help to Select the Most Relevant Predictors?

Including disaggregate variables or using information extracted from the disaggregate variables into a forecasting model for an economic aggregate may improve forecasting accuracy. In this paper we suggest using the boosting method to select the disaggregate variables, which are most helpful in predicting an aggregate of interest. We conduct a simulation study to investigate the variable selection ability of this method. To assess the forecasting performance a recursive pseudo‐out‐of‐sample forecasting experiment for six key euro area macroeconomic variables is conducted. The results suggest that using boosting to select relevant predictors is a feasible and competitive approach in forecasting an aggregate. Copyright © 2016 John Wiley & Sons, Ltd.     

Forecast intervals of net migration: the case of the Netherlands

This study addresses problems concerning the forecasting of net migration in the preparation of population forecasts. "As the width of forecast intervals for migration in single years differs strongly from that of an interval for average migration during the forecast period, it is important that the forecaster indicates which type of interval is presented. A comparison of forecast intervals for net migration obtained from an ARIMA model to intervals in official Dutch national population forecasts shows that the uncertainty on migration has been underestimated in past official forecasts."     

Forecasting key macroeconomic variables from a large number of predictors: a state space approach

We use state space methods to estimate a large dynamic factor model for the Norwegian economy involving 93 variables for 1978Q2–2005Q4. The model is used to obtain forecasts for 22 key variables that can be derived from the original variables by aggregation. To investigate the potential gain in using such a large information set, we compare the forecasting properties of the dynamic factor model with those of univariate benchmark models. We find that there is an overall gain in using the dynamic factor model, but that the gain is notable only for a few of the key variables. Copyright © 2009 John Wiley & Sons, Ltd.