Research Discussion Paper – RDP 2024-04 Nowcasting Quarterly GDP Growth during the COVID-19 Crisis Using a Monthly Activity Indicator

What is happening in the economy now? It is said that the future is uncertain, but so is the present. Policymakers operating in this environment need some way to understand what is happening now (i.e. current economic conditions). This need for timely information was most acute during the COVID-19 crisis when current conditions were rapidly evolving, requiring policymakers to make decisions under significant economic uncertainty. Policymakers are further hamstrung because the most comprehensive measure of economy activity, gross domestic product (GDP), is published with a substantial lag. Indeed, the full effect on economic activity of the first major lockdowns which occurred during June quarter 2020 were not realised until the release of National Accounts data in early September 2020; more than two months after the reference period.^[1] This delay limits its value to policymakers as a measure of the current state of the economy. Additionally, GDP is often revised in subsequent quarters which further limits its usefulness to policymakers for assessing current conditions.

In response, more and more higher-frequency partial indicators have become available in recent times; however, they are often not as comprehensive in their scope and coverage as traditional measures of economic activity such as GDP. And while these partial indicators do help fill the information gap, the signal they provide is often noisy. Further, one indicator might be useful in one context but not in another. For example, the unemployment rate is considered a key metric of economic activity, but during the COVID-19 crisis the Australian Government introduced the ‘JobKeeper’ program to keep workers employed, thereby limiting the rise in the unemployment rate caused by lockdowns.^[2] During this period, the underemployment rate was considered to provide a more accurate signal. Given this, it is not clear how policymakers should choose which indicator to focus on, and if there are multiple indicators available, how much weight they should give to each one. The answers to these decisions are subjective and will typically vary with time and across policymaker.

What is required is a method of combining the available partial indicators in a systematic manner to smooth out the noise and reveal the underlying signal. The most common tool for achieving this is via dynamic factor models (DFMs). DFMs are a dimension reduction technique that can summarise the common variation across a panel of time series data.^[3] In Australia, initial work exploring the usefulness of factor models for monitoring economic activity was undertaken by Gillitzer, Kearns and Richards (2005). They produced two coincident indicators, one summarising quarterly data and another summarising monthly data. Both indicators were estimated using the non-parametric methods developed by Stock and Watson (2002) and Forni et al (2000).^[4] This was followed by Sheen, Trück and Wang (2015), who introduced a daily business cycle indicator based on the work of Aruoba et al (2009). Their method uses a parametric estimation technique involving a state-space model estimated using the Kalman filter.

Besides being a successful tool for monitoring activity, another important use of DFMs is for prediction (Stock and Watson 2002), especially in relation to producing nowcasts.^[5] A significant amount of research effort has been devoted to this topic since the early works of Nunes (2005) and Giannone, Reichlin and Small (2008) (see Bańbura et al (2013)). However, there has been considerably less work done in Australia. For prediction, Gillitzer and Kearns (2007) had success, showing factor-based forecasts for key macroeconomic series can outperform standard time series benchmarks.^[6] The benefits of DFMs are less clear when focusing explicitly on nowcasting in Australia (Australian Treasury 2018; Panagiotelis et al 2019). Using different estimation techniques both suggest the sample mean is a difficult benchmark model to beat in relation to nowcasting quarterly GDP growth.^[7] But, while both works consider higher frequency data (monthly and higher), neither exploit this information in their nowcasting investigation. Instead, both convert all series in their respective datasets to a quarterly frequency before producing a nowcast.^[8],[9] This is a problem because it represents a loss of information. Further, there is extensive research highlighting the significant improvement in prediction accuracy that comes from working with mixed frequency data.^[10]

Our work bridges the gap that exists in the literature between employing factor models for monitoring and for nowcasting in Australia. Both issues are interrelated and are equally important for policymakers, so it is sensible to develop a framework that can achieve both objectives at once. In doing so, we build on previous work in Australia by incorporating more recent developments in factor modelling and nowcasting. We start by developing a monthly activity indicator (MAI) for Australia. The MAI aims to provide policymakers with a more immediate snapshot of prevailing economic conditions. We achieve this by using a ‘true’ DFM to summarise the information content from a dataset of 30 monthly targeted predictors selected for their ability to explain movements in first-release quarterly GDP growth.^[11] This is an important advance compared to previous studies as it links the variable of interest to the estimation of the DFM and has been shown to improve factor estimation and predictive ability (Bai and Ng 2008; Bulligan, Marcellino and Venditti 2015). We also extend the targeted predictor hard thresholding pre-selection step developed by Bai and Ng (2008) when estimating factor models to the mixed frequency setting. Further, the methodology we use to estimate the MAI allows us to use an unbalanced dataset which therefore means we can consider a broader collection of series over a longer time period than the competing indicators produced by Gillitzer et al (2005) and Sheen et al (2015).

Unlike previous nowcasting studies in Australia (e.g. Australian Treasury 2018; Panagiotelis et al 2019), which have focused exclusively on quarterly frequency data, we undertake the first investigation of nowcasting in Australia using a mixed frequency modelling framework. We exploit the MAI's high-frequency information content within a factor augmented unrestricted MIDAS (MIxed Data Sampling) model (FA-U-MIDAS).^[12] We assess the model's ability to nowcast first-release quarterly GDP growth using a recursive out-of-sample evaluation exercise covering a 34-year period (1988:Q2–2022:Q2), longer than previous works including Gillitzer and Kearns (2007), Australian Treasury (2018), and Panagiotelis et al (2019). Additionally, since we use monthly data, we can generate four nowcasts for each quarterly GDP growth observation as new monthly data becomes available across the quarter. Finally, as in previous evaluations, we use the standard benchmark forecasting models of the sample mean and an AR(1) process for comparison.

Our results show that incorporating monthly information provides more accurate predictions compared to the benchmark models based on smaller estimated root mean squared error. The improvement over the benchmark models (sample mean and AR(1) models) is also found to be statistically significant as well.^[13] Crucially, predictive accuracy of the models with monthly data is largest during the COVID-19 crisis compared to the benchmark models relying solely on quarterly data, highlighting the benefit to policymakers from having timely information. Our results also support previous findings which suggest that model predictive performance can change depending on the state of the economy (see Chauvet and Potter (2013), Siliverstovs (2020) and Jardet and Meunier (2022)).

We begin by describing in detail the methods we follow to construct the MAI in Section 2. In Section 3 we discuss how we use the MAI to predict quarterly GDP growth as well as the steps we follow to implement the out-of-sample evaluation exercise before concluding in Section 4. Some additional results are provided in the appendices.

To construct the MAI, we apply a DFM to a monthly dataset. The dataset will be comprised of series which show a statistically significant relationship with quarterly GDP growth since this is our variable of ultimate interest. We first discuss the monthly dataset and how we select the series it contains before describing the DFM method and estimation results.

2.1 Monthly activity dataset

The choice of dataset to use when estimating a DFM is an important part of the process that is often underappreciated. There no single agreed upon way to do this in the literature. Indeed, different datasets can result in different factor estimates even when using the same estimation technique (Bai and Ng 2008). Given this, there is a tendency for researchers to select as many series as possible in the attempt to capture all available information. However, having too many series can be problematic for factor estimation as well; especially if many of the series in the dataset are ‘noisy’ (Boivin and Ng 2006).^[14]

Relatedly, other methods for creating datasets from pre-selected series have been proposed to generate more accurate forecasts with factor-augmented regressions. Unlike those proposed by Boivin and Ng (2006), these methods recommend estimating a factor model using a dataset comprised of only those series shown to have predictive power for a ‘target’ variable of interest. Importantly, these so-called ‘targeted predictors’ explicitly take account of the object of interest which other methods do not. Two prominent strategies include ‘hard’ and ‘soft’ thresholding to determine which variables the factors are to be extracted from (Bai and Ng 2008). Under hard thresholding, the predictors are ranked based on a pre-test procedure and those that fail to meet some criteria are discarded from the dataset. Under soft thresholding, a portion of top ranked predictors are kept, where the ordering of the predictors depends on the soft thresholding rule used. Bai and Ng (2008) show that factors extracted from a dataset of targeted predictors can result in superior forecasting performance.^[15]

To begin, we compile an ‘extended’ dataset that includes 53 monthly partial indicators covering various aspects of the Australian economy. Following Bańbura and Rünstler (2007) we group these series into three main categories: ‘hard’ (30 per cent; includes series covering key measures of activity such as the labour market); ‘soft’ (36 per cent; includes survey measures which tend to be more timely than hard series); and ‘financial’ (34 per cent; includes series such as interest rates, equity prices and commodity prices). When available, we include both aggregate and disaggregate measures in the dataset (i.e. total credit as well as its sub-components). Some researchers argue against this practice; however, the method we use to estimate the DFM is robust to including aggregate and disaggregate series.^[16] The dataset covers the sample period 1978:M2 to 2022:M9 and was influenced by the number of series available in the early part of the sample.^[17] However, several series in the dataset have later starting and earlier ending periods due to being relatively new, so the resulting dataset is ‘unbalanced’ or ‘ragged edge’.

Before using the dataset, we transform all series to be stationary and standardise them to have zero mean and unit variance as is common in the factor modelling literature. Series are made stationary by taking logs and/or first differences as appropriate (see Table A1 for details). When doing the standardisation, rather than use the full sample mean, we instead follow Kamber, Morley and Wong (2018) and implement ‘dynamic demeaning’ for each series using a rolling 20-year backward-looking estimate of the sample mean as a way of controlling for potential structural breaks in the central tendency of each series over the sample period the dataset covers. The decision to use a 20-year window (instead of a 10-year window as in Kamber et al (2018)) is because until the COVID-19-induced recession in 2020, the length of the business cycle in Australia was arguably longer than elsewhere.^[18]

Since our main goal is to produce a monthly activity indicator for monitoring the economy at a higher frequency than is currently possible and to predict quarterly GDP growth in the near term, we will follow Bai and Ng (2008) and implement a pre-selection strategy to our extended dataset to remove any uninformative predictors in relation to quarterly GDP growth. Because our dataset is unbalanced, we will use their hard thresholding strategy. This involves running a series of separate regressions of the target on a single predictor. Each regression includes a set of controls comprised of an intercept and lags of the target variable which are the same for all regressions. The predictors are then ranked in descending order by the magnitude of the coefficient t-statistic on each predictor. Any predictor with a test statistic below some specified threshold significance level is discarded.^[19],[20]

The method we use to estimate the DFM is robust to model misspecification. Hence, one could argue there is no need to apply any pre-selection to the dataset since the model will assign the right weight to each series (see Bańbura et al (2013)). However, factors extracted from the extended dataset will, by construction, be a linear combination of all series in the dataset. Some of these series might not be very informative about quarterly GDP growth but will still have some effect on the model outputs even if small. That is, no series is likely to be assigned a zero weighting. Therefore, it makes sense to only focus on a subset of series found to be informative about the quarterly growth in GDP.^[21]

Instead of using the current release version of GDP, which is a combination of first release, revised and fully revised data (Stone and Wardrop 2002), we follow Koenig, Dolmas and Piger (2003)'s recommendation and use the first-release version of GDP (Lee et al 2012). We extend Bai and Ng (2008)'s hard-thresholding algorithm (which only considers variables at a quarterly frequency) to a mixed frequency setting. This is because our target variable is quarterly while our predictors are monthly. In this situation, it is typical to perform some type of temporal aggregation such as taking the quarter average (i.e. each quarterly observation is the average of the three monthly observations in each quarter). However, this could result in a potential loss of information. Instead, each monthly series is converted to a quarterly series by stacking the first, second and third months in each quarter as three separate quarterly series.^[22]

Because we have three predictors (i.e. one series each for the first, second and third months of the quarter) instead of only one as in Bai and Ng (2008) and Bulligan et al (2015), we cannot implement the same t-statistic to test for significance and rank series as they both do. Instead, we test for the joint (linear) significance for all three series at once using a Wald statistic calculated using a HAC robust covariance matrix. As controls we include an intercept and, as our sample covers the COVID-19 crisis period, a set of seven indicator variables for the periods 2020:Q2 to 2021:Q2 and 2021:Q4.^[23] The indicator variables were included to account for the COVID-19 crisis so as not to affect the test results and series ranking.^[24] When running each regression, we adjust the dependent variable's sample length to match the sample length of each predictor which varies by series. Because our extended dataset is already relatively small by international standards, we use a less restrictive significance level of 10 per cent to gauge significance.^[25] While this is higher than the standard 5 per cent, it helps ensure we have a reasonable sized subset of the extended dataset.

The outcome from the hard thresholding procedure is a dataset of 30 variables from the original 53-variable extended dataset.^[26] Of the three categories, ‘soft’ is the dominant one with 13 series (43 per cent); followed by ‘financial’ with 9 (30 per cent) and ‘hard’ with 8 (27 per cent). Figure A1 shows the 30 series by category and ranked by Wald statistic along with the threshold critical value (dashed line). The number of series in the targeted predictor dataset is comparable to the minimum suggested by Bai and Ng (2008) and is slightly larger than the 24 series used in the empirical application by Bańbura et al (2013) and slightly smaller than the 37 series used by Australian Treasury (2018). Further, Panagiotelis et al (2019) mention they find no benefit from considering an information set bigger than 20 to 40 variables when forecasting Australian macroeconomic time series such as quarterly GDP growth.^[27]

2.2 Constructing the monthly activity indicator using a dynamic factor model

DFMs are a popular statistical model for summarising the common (linear) variation contained in a panel of time series data and prediction. A key issue of all these previous works is that they not true DFMs as per Bai and Wang (2015). Instead, we estimate the MAI using the general form of the DFM defined as:

where y_t is a N × 1 vector of weakly stationary targeted predictors, f_t is a q × 1 vector of the dynamic factors, and ${\Lambda }_i$ is the dynamic factor loadings for f_t–i with i = 0, 1,...,s and t = 1,...,T. Together, the factors and loadings provide a measure of the common variation shared across series in the dataset. The dynamic factors are modelled as a VAR(p) process with ${\Phi }_i$ a q × q matrix of autoregressive coefficients (with all roots outside the unit circle). The number of dynamic factors is q (the dimension of f_t).

The covariance matrix of the idiosyncratic component ${\epsilon }_t$ is given by R with dimension N × N and is restricted to be a diagonal matrix. In the state equation, the covariance matrix of ${\eta }_t$ corresponds to the q × q matrix Q . We assume that $E\left[{\epsilon }_t{{\eta }^{\prime }}_t\right]=0$ (i.e. the two noise processes are independent). This specification of the DFM has two different sources of dynamics. First, there are s lagged factors representing a dynamic relationship between the observable series y_t and the factors f_t. Second, the dynamics of the factors are assumed to be captured by a VAR(p) process.^[28] Bai and Wang (2015) argue that it is the first source of dynamics that makes this specification a true dynamic factor model because it is these dynamics that make the biggest distinction between dynamic and static factor analysis.^[29]

We estimate the DFM by quasi-maximum likelihood (QMLE).^[30] Estimation is conducted via the expectation-maximisation (EM) algorithm and consists of two parts. First, we estimate the factors given the data by running the Kalman filter and Rauch-Tung-Striebel (RTS) smoother recursions (the ‘E-step’). Second, we use the estimated factors from the previous step to compute the model parameters by maximising the expected log-likelihood by regression (the ‘M-step’). This requires us to re-cast Equation (1) into its state-space representation given as:

The measurement equation takes the form of a static factor model (Stock and Watson 2002) with r = q (s + 1) static factors. Let k = max(p, s + 1), then F_t is a qk × 1 vector of the dynamic factors and their lags, $\Lambda$ is a N × qk matrix of dynamic factor loadings, $\Phi$ is a qk × qk companion matrix and G is a qk × q selector matrix. The advantage of using the state-space modelling framework is that it can easily and efficiently accommodate unbalanced datasets. See Hartigan and Wright (2023) for more details on the parameters and estimation procedure we use.

2.2.1 Determining the optimal DFM specification

Before we can estimate the DFM we first need to specify four important features. These are: i) the number of dynamic factors ( q ), ii) the number of dynamic loadings ( s ), iii) the lag order for the factor VAR in the state equation ( p ), and iv) the ‘named factor’ necessary for identification.

We use the information criterion developed by Hallin and Liška (2007) to determine the number of dynamic factors. This suggests there is only one common dynamic factor in the targeted predictor dataset (see Figure A4).^[31] To set the number of dynamic factor loadings, we follow the strategy implemented in Luciani (2020). This exploits the fact that a dynamic factor model with q factors can be re-cast as a static factor model with r = q (s + 1) static factors as previously mentioned. Practically, we take a balanced subset of the targeted predictor dataset and compare the proportion of explained variation from the first r eigenvalues from the contemporaneous covariance matrix to the proportion of variation from the first q dynamic eigenvalues from the spectral density matrix averaged over a grid of frequencies (see Forni et al (2000) and Brillinger (1981) for more details). The aim here is to find where there is close agreement between these two measures. Examining Table 1 indicates that one dynamic eigenvalue (i.e. q = 1) explains approximately the same amount of variation as three static eigenvalues (i.e. r = 3) and hence suggests s ≈ 2. Further, Luciani (2020) argues that with s = 2 each series in the targeted predictor dataset is capable of loading on the dynamic factor in a time window of three months. This is interesting as this window corresponds to one quarter.

We can check whether Boivin and Ng (2006)'s suggestion that reducing the sample size can ‘sharpen the factor structure’ by comparing the amount of explained variation from the extended dataset and targeted predictor dataset. Focusing only on the first dynamic eigenvalue, the amount of variation explained in a balanced subset of the extended dataset is about 52 per cent (not shown), lower than the per cent of explained variation in the pre-screened dataset (Table 1). Hence, removing any series considered uninformative in relation to explaining movements in quarterly GDP growth has increased the signal-to-noise ratio of the common dynamic factor.

With only one common factor, the dynamics of the factor follow an AR process instead of a VAR process. To determine the lag order of the AR process we set this to one (i.e. p = 1) based on the AIC.

The final task needed before estimation can take place is identification. To identify the DFM we impose the ‘named factor’ normalisation (Stock and Watson 2016), which associates a factor with a specific variable.^[32] In deciding which targeted predictor to make the named factor we put ‘WMI consumer sentiment’ as the first series because this series has the highest Wald statistic of all the 30 targeted predictors (see Figure Al).

2.2.2 Estimation results

Figure 1 presents the (optimal) filtered estimate of the MAI for the sample period: 1978:M2 to 2022:M9.^[33] The MAI reveals three periods of relatively weak activity that correspond with previous recessions in Australia, with the most recent being from the COVID-19 crisis (i.e. 1982, 1989-1991 and 2020). Indeed, the decline in the level of the MAI during this period is the largest ever observed in the series. Although the duration was much shorter than compared to the other two time periods and is focused predominately in June 2020.

Note: Shaded bars are Sahm rule recession onset dates for Australia, defined as a ¾ percentage point increase in the three-month moving average of the unemployment rate relative to its minimum in the previous twelve months.

The MAI also shows activity was noticeably weak in two other periods which have not previously been considered as recessions as per the technical definition of a recession. The first period, 2001, is linked to the aftereffects of the introduction of the GST which caused a significant amount of activity to be brought forward. The second period, 2008, corresponds with the global financial crisis (GFC). However, both periods together with the three previously acknowledged recessions are detected by the so called ‘Sahm rule’ (Sahm 2019). This is an algorithm for detecting the onset of recessions based on monthly movements in unemployment and has correctly detected every recession in the United States since the 1970s as identified by the NBER, with no false positives.^[34] We adjust the Sahm rule for Australia and consider a ¾ percentage point increase in the three-month moving average of the first-release unemployment rate relative to its minimum during the previous twelve months to be more appropriate. With no widely recognised recession timing for Australia equivalent to the NBER Business Cycle Dating Committee for the United States the Sahm rule serves as a useful proxy (see He and Rosewall (2020)). Figure 1 also shows that downturns in the MAI appear to occur several months before detection by the Sahm Rule. In this sense, the MAI provides an important signalling device for policymakers related to probable downturns.

To understand movements in the MAI over time we need to quantify the contributions of individual series. This is something not directly possible to do via non-parametric techniques such as PCA. These contributions are not provided as part of the estimation procedure, but they can be obtained from the state-space representation of the model in Equation (2). First, we take the state equation part of the model and re-write this expression in terms of the updating equation from the Kalman filter:

where K_t is the Kalman Gain at time t and is of dimension r × N and the other parameters are as previously defined. Equation (3) says that the estimate of the common factor at time t is a linear combination of a prediction step (based on information at t–1) and an update step based on the error in the prediction weighted by the Kalman Gain. This second part gives us the contribution from each series to the factor at each time point (see also Sheen et al (2015)).

Next, let D_t denote a r × N matrix of series-specific contributions and using Equation (3) we can get an expression for the update step for each series:^[35]

Equation (4) is related to Equation (3) by noting that $F_t=\Phi F_{t-1}+D_t{\iota }_N$ where ${\iota }_m$ is a column vector of 1s with the number of elements specified by m. The first row of D_t for t = 1,...,T gives the individual series-specific contribution to the common dynamic factor f_t as defined in Equation (1). Figure 2 plots the contributions to the Kalman filter estimate of the MAI aggregated by data category (i.e. hard, soft or financial) for the period January 2000 until July 2022 to allow for a more easy interpretation of recent history.

Note: Contribution to state update in Kalman filter recursions.

Figure 2 reveals the soft data category is the main contributor to updates in the MAI followed by hard and financial data. What is interesting about this observation is that the GFC was typically thought of as a financial crisis. However, the break down of the MAI by data category in Figure 2 reveals the previously discussed weakness in the MAI that occurred during the GFC in Australia was primarily due to a decline in soft data and these are mostly sentiment-based series.^[36] Financial-based series only contributed a very small amount during that period.

This makes sense because during the GFC the economic environment was very uncertain and there was a lot of pessimism expressed by both consumers and businesses. However, fears of a serious recession turned out to be premature due to a combination of a very large fiscal response by the Australian Government, a very aggressive loosening of monetary policy by the RBA and a surge in demand for commodities from China. The steady rise observed in the MAI until early 2018 was also predominately caused by soft data. More recently, the dramatic movements observed in the MAI during the COVID-19 crisis period were due to contributions from both the hard and soft data categories, with the financial data category only making a relatively minor contribution.

This analysis reveals a potential issue that users of the MAI as a measure of activity need to consider. While soft data, such as surveys, do have the advantage of being very timely compared to hard data categories, they can also provide false signals (Aylmer and Gill 2003). Further, Roberts and Simon (2001) conclude that the information content that survey data, such as sentiment indicators, does provide is at best only a rough summary of prevailing economic conditions. However, they note that in some cases a linear combination of survey indicators (as is the case with a DFM) might not be a bad compromise.

As previously stated, the DFM we use to construct the MAI has been shown to be robust to misspecification including conditional heteroskedasticity and ‘fat tails’ (i.e. outliers) when the factors are extracted from many variables (see Doz et al (2012) and Bańbura et al (2013)). However, it is evident from Figure 1 that the COVID-19 crisis had a substantial effect unlike anything observed before on many of the series included in the targeted predictor dataset. Further, Maroz, Stock and Watson (2021) document how the COVID-19 crisis resulted in a temporarily large change in previously observed patterns of co-movement across a panel of US monthly time series data. While they use a different model than we do, it is still important to check the robustness of our model estimation.

The way we do this in our work is to compare two versions of the MAI constructed using parameters estimated from the full sample (including the COVID-19 crisis, labelled ‘FS’) and parameters estimated up to 2020:M2 (i.e. the pre-COVID-19 crisis, labelled ‘PC’). The results are illustrated in Figure 3 (upper panel) while the difference between the two MAI estimates is displayed in the lower panel. Visually, both MAI estimates look broadly similar. The main difference is that the PC estimate does not fall as dramatically during the worst of the COVID-19 crisis in June 2020. The sample standard deviation of the difference measure for the full sample is 0.15, while the sample standard deviation of the difference measure for the pre-COVID-19 sample is 0.11. The null hypothesis that the sample standard deviation of the full sample difference measure cannot be rejected at standard levels of significance.^[37]

One reason for the smaller observed effect in our case compared to the findings of Maroz et al (2021) could be because our dataset does not display the same extreme movements during the COVID-19 crisis as their dataset. Indeed, they report one series having declined by more than 275 standard deviations. In our dataset, the largest decline was much smaller (see Figure A6). Further, given the relatively smaller decline observed in the MAI (PC) during the COVID-19 period, it is reasonable to argue that we need to include the COVID-19 period to ensure we correctly estimate its effect across series and the economy when we turn to nowcasting quarterly GDP growth in the next section.

Note: ‘FS’ refers to MAI based on parameters estimated using the full sample; ‘PC’ refers to MAI based on parameters estimated using a sample ending 2020:M2 (i.e. pre-COVID-19).

In this section we take the estimated MAI from the previous section and develop a framework for nowcasting quarterly growth in GDP. To achieve this goal, we will develop a regression model that relates the MAI to movements in quarterly GDP growth. This will require us to work with mixed frequency data.

3.1 Modelling mixed frequency data

When modelling time series of different frequencies, the typical thing to do is convert all time series to the lowest observed frequency using temporal aggregation. Usually this involves computing the average of the observations of the high-frequency variable that occur between samples of the low-frequency variable. For example, with monthly/quarterly data this could involve taking the average of the three months in the quarter or the last monthly observation in the quarter. The former was the approach adopted by previous studies in Australia including Gillitzer and Kearns (2007), Australian Treasury (2018), Panagiotelis et al (2019).^[38] And while it is simple to implement, it discards potentially important information about the timing of movements in the high-frequency variable. Indeed, the reason we developed the MAI was so we could exploit the timely information it provides.

Instead, we will employ the MIDAS regression modelling framework (see Ghysels et al (2004) and Ghysels et al (2007)). MIDAS regression provides a flexible way to directly exploit all the information content of a higher-frequency explanatory variable to predict a lower-frequency dependent variable. It achieves this by using highly parsimonious distributed lag polynomials to prevent parameter proliferation that might otherwise occur.^[39] MIDAS regression has been successfully used for predicting macroeconomic and financial variables. Of relevance to our work, Clements and Galvão (2008) show that using monthly information on the current quarter leads to significant improvements in forecasts based on coincident indicators.

An alternative approach that other researchers have used to handle mixed frequency data is to specify a state-space model and estimate it using the Kalman filter (e.g. Bok et al 2017).^[40] However, in comparison with MIDAS models no clear ranking of forecast performance between the two methods was found (Bai, Ghysels and Wright 2013). Overall, Bai et al conclude that MIDAS and state-space models give similar forecasts.^[41],[42]

The simple MIDAS model incorporating a single regressor is given by:

where $W\left(L^{1/m};\theta \right)={\Sigma }_{k-0}^{K-1}W\left(k;\theta \right)L^{k/m}$ and L^1/m is a high-frequency lag operator such that $L^{1/m}{x}_t^{\left(m\right)}=x_{t-1/m}^{\left(m\right)}$ with m indicating the higher sampling frequency of the explanatory variable (for example, m = 3 when x is monthly and y is quarterly). The intercept is specified by ${\beta }_0$ while the coefficient ${\beta }_1$ captures the overall effect of the high-frequency variable x on y and can be identified by normalising the function $W\left(L^{1/m};\theta \right)$ to sum to one. We assume the residuals, ${\in }_t$ are an iid sequence with mean zero and constant variance. Finally, K is the maximum lag length for the included high-frequency regressor.

The method by which the MIDAS model achieves a parsimonious representation is via the lag coefficients in $W\left(k;\theta \right)$. This represents a set of weights as a function of a small dimensional vector of j parameters $\theta =\left({\theta }_0,{\theta }_1,\mathrm{...},{\theta }_j\right)$ with $j\ll K$. Two common functions used in empirical applications include the normalised exponential Almon lag function of Ghysels et al (2004) given as:

And the normalised beta function of Ghysels et al (2007) given as:

Note, Figure B1 illustrates examples of both polynomial weighting functions for different sets of parameters. Since both weighting functions are highly nonlinear, MIDAS models featuring either of them will need to be estimated by nonlinear least squares. An alternative specification proposed by Foroni et al (2015) is ‘unrestricted MIDAS’ (U-MIDAS). This method leaves the high-frequency lag coefficients unconstrained and can be estimated by OLS.^[43] Foroni et al (2015) show that U-MIDAS is often preferable to standard (i.e. restricted) MIDAS (R-MIDAS) when modelling quarterly and monthly data because m is small. This reflects the fact that when the number of lags to model is relatively small, complications caused from having to estimate more parameters are reduced.^[44] A U-MIDAS model with one explanatory variable is given as:

where $B\left(L^{1/m}\right)={\Sigma }_{k=0}^{K-1}{\beta }_k{L}^{k/m}$ and L^1/m is defined as before. In any case, our x variable (i.e. the MAI) will be a latent common dynamic factor. MIDAS models incorporating factors are typically referred to as factor augmented MIDAS (FA-MIDAS) models and have been shown to perform well compared to more standard quarterly factor models in short-term forecasting of quarterly GDP growth in Germany (see Marcellino and Schumacher (2010)).^[45] This finding is important for our work in two ways. It confirms the benefit to prediction from using mixed-frequency techniques and suggests that FA-MIDAS models can exploit time series information more efficiently than existing approaches.

Before we can move onto specifying a MIDAS model for nowcasting quarterly GDP growth using the MAI we need to decide on two aspects about the specification we intend to use. First, the functional constraints (if any) to implement and second, the optimal maximum lag order K. One way to address both issues is to use an information criterion to select the best model in terms of parameter restriction and the lag orders based on in-sample model fit.

Since the selected model will ultimately be used for nowcasting, we follow standard practice in the forecasting literature and use real-time data.^[46],[47] For the dependent variable y we use first-release GDP from Lee et al (2012).^[48] The main argument for this decision is that data revisions to GDP cause an additional issue when nowcasting. If we focus on current GDP, which is a combination of first release, partially revised and fully revised data, then we not only have to consider how to nowcast the first release of quarterly GDP growth but also how to predict future data revisions. Further, revisions to GDP can occur many quarters after the initial release. So, it is reasonable to assume analysts are more interested in the initial releases and concerned with the uncertainty related to nowcasting the first release than the uncertainty related to the revision process (Galvão and Lopresto 2020).

Unlike with GDP, there is no vintage targeted predictor dataset available for constructing a genuine real-time version of the MAI. However, as an alternative, we use the estimate of MAI produced by the Kalman filter for the reason previously discussed about it being more appropriate for prediction since it only incorporates information up to time t. Further, it is also conceptually similar to the definition of a real-time variable provided by Koenig et al (2003).^[49]

We follow Foroni et al (2015) and use the BIC to evaluate a range of restricted and unrestricted MIDAS models. For the R-MIDAS models, we consider the normalised exponential Almon weighting function with j = 2 and j = 3 parameters. We also consider the normalised beta weighting function with j = 3 parameters. For all MIDAS model specifications, we specify four values for the maximum lag of the monthly explanatory variable (i.e. K $\in \left\{2,3,4,5\right\}$ ).^[50] The results are presented in Table 2. The BIC strongly prefers the U-MIDAS specification with maximum lag K = 5, this is closely followed by the U-MIDAS model with maximum lag K = 6. Indeed, all U-MIDAS models are superior to the two R-MIDAS models except for when the maximum lag is two (K = 2). In this case, the R-MIDAS model using the normalised exponential Almon polynomial weighting function with two parameters (j = 2) is preferred.

In addition to comparing models based on the BIC, it is also possible to test the empirical adequacy of the polynomial weighting functions used with the R-MIDAS specifications under standard assumptions via a Wald-type test. The null hypothesis is that the functional restrictions are valid. Therefore, rejecting the null implies the functional restrictions are not supported by the data. By this metric, only one R-MIDAS model specification is consistent with the data, corresponding to the normalised exponential Almon polynomial weighting function with K = 2 and j = 2.

3.2 Out-of-sample prediction comparison

In this section we will assess the nowcasting performance of MIDAS models incorporating the MAI compared to standard benchmark models in a pseudo out-of-sample (OOS) comparison exercise. Based on the findings of the model evaluations presented in Table 2 we will only consider the FA-U-MIDAS specification for nowcasting quarterly GDP growth. However, instead of setting K = 5 as suggested by the BIC, we set K = 6. This choice is motivated by previous work (see Koening et al (2003) and Leboeuf and Morel (2014)) and because it covers the months most likely to affect quarterly GDP growth (i.e. three months of data covering the quarter for which we observe the last value of real GDP growth and the three months of data covering the first quarter to nowcast).^[51]

We do not consider longer horizon predictions of quarterly GDP growth as in other studies. This is because predicting output growth over longer horizons is known to be much less reliable (e.g. Marcellino and Schumacher (2010) for FA-MIDAS models, Bańbura et al (2013) for factor models, and Chauvet and Potter (2013) for a systematic evaluation more generally). As such, the methods we develop here should only be thought of as short-term prediction devices.

An important advantage of MIDAS regression over other methods used for handling mixed frequency data (i.e. temporal aggregation) is that it allows us to make predictions within periods. Further, each successive prediction will incorporate a new estimate of the MAI as more data becomes available in the quarter. For simplicity, we assume the following timing of data releases. Let y_t denote the current quarter of quarterly GDP growth and y_t+1 denote the next quarter of quarterly GDP growth. The first release of GDP for quarter t contains data up to quarter t – 1. Before data on GDP growth for t become available in quarter t +1, we will have four updates of the MAI. First estimate of the MAI incorporating monthly data up to t –1 in t –2/3 (i.e. first month of current quarter), second estimate of the MAI incorporating monthly data up to t –2/3 in t –1/3 (i.e. second month of current quarter), third estimate of the MAI incorporating monthly data up to t –1/ 3 in t (i.e. end of the current quarter). Finally, a fourth estimate of the MAI incorporating monthly data up to t in t +1/3 (i.e. first month of the next quarter). The timing of these monthly updates of the MAI allows us to produce four predictions of quarterly GDP growth in the current quarter which we label as i) forecast (FC), ii) nowcast in month 1 (M1), iii) nowcast in month 2 (M2), and iv) nowcast in month 3 (M3).^[52] See Figure 4 for a visual summary.

Based on this, the general FA-U-MIDAS model we use in the OOS evaluation becomes:

where y_t is first-release quarterly GDP growth, x_t is the MAI and $i\in \left\{3,2,1,10\right\}$ depending on the monthly flow of data during the quarter (i.e. corresponding to the four predictions: FC, M1, M2, M3, in that order). Hence, as new monthly estimates of the MAI are produced during the quarter, the specification of the FA-U-MIDAS model will change with an increasing number of regressors. For example, when i =3 (i.e. FC), the FA-U-MIDAS model for current quarterly GDP growth consists of an intercept and three months of data on the MAI from the previous quarter:

Alternatively, when i =0 , the model expands to include three additional months of data on the MAI from the current quarter (reflecting the full model):

Note, for ease of notation, we drop the superscript (m) from the x variable in both equations. Across the OOS evaluation period we will keep the FA-U-MIDAS model specification fixed to this general form.^[53] Further, we will compare the four FA-U-MIDAS model specifications to two standard models used in previous OOS forecasting/nowcasting evaluation exercises. These include the sample mean and an AR(1) process (see Australian Treasury (2018) and Panagiotelis et al (2019) for the sample mean and Gillitzer and Kearns (2007) for the AR(1) process). The sample mean has been shown to be a formidable forecasting model for quarterly growth in GDP (Panagiotelis et al 2019) and will serve as our benchmark model in our comparisons. Additionally, we also consider a model based on a quarter average (QA) measure of the MIA as a crosscheck.^[54] The QA model includes a temporal aggregated value of the MAI for the current quarter and another lagged value for the previous quarter, making it similar to M3.

To evaluate the performance of the various models, we carry out a recursive estimation and forecast/nowcasting exercise, where the full sample is split into estimation and evaluation sub-samples. The estimation sample initially covers the period 1978:Q2–1988:Q1 (i.e. R = 40 or 10 years, similar to Panagiotelis et al (2019)) and is expanded by one quarter at a time and the model parameters are re-estimated each time. The evaluation sample is between 1988:Q2 and 2022:Q2 (i.e. P =137 ). For each quarter in the evaluation sample, we want to compute a forecast and three nowcasts depending on the monthly information set. For example, for the initial evaluation quarter 1988:Q2, we want to compute a forecast using data up to 1988:Q1 (FC) and then a nowcast in 1988:M4 (M1), 1988:M5 (M2) and 1988:Q2 (M3). At this point we also compute the sample mean and AR(1) forecasts. The predictions from each model over the evaluation sample are presented in Figure 5.

Note: GDP is first release.

Sources: ABS; Authors' calculations; Lee et al (2012).

The sample mean and AR(1) predictions are very similar given the lack of persistence in quarterly GDP growth, although the AR(1) model was slightly better at predicting the COVID-19 decline in 2020:Q2, albeit with a quarter lag. Since both models are estimated using quarterly data, neither were able to fully anticipate the significant fall and immediate rise that eventuated in 2020:Q2 and 2020:Q3. In contrast, the models incorporating monthly information performed much better. Predictions from each model are reasonably similar in the period before the COVID-19 crisis, but all models show noticeable differences in the period afterwards. For example, model M1 was most accurate in predicting the contraction in quarterly GDP growth in 2020:Q2, although it was still off by around 2 percentage points. Alternatively, model M3 was relatively less successful. This is surprising, since model M3 has two extra months of data on 2020:Q2 and previous research has shown that having more timely data usually improves forecast accuracy.^[55] One explanation could be that model M3 has two additional parameters to estimate compared to model M1 and increased estimation uncertainty could be affecting the model's accuracy.

In relation to predicting the large subsequent upswing in quarterly GDP growth in 2020:Q3, more success was achieved by models FC and QA. This is also surprising since both models contain less information on the quarter compared to the three ‘M’ models. However, both models have less parameters to estimate compared to the other models (FC with three and QA with only two) and therefore could be more precisely estimated, improving the accuracy of both models.

We assess point forecast/nowcast accuracy of each model considered using standard root mean squared error (RMSE) defined as:

where ${\widehat{y}}_t$ is the forecast/nowcast produced by one of the models and P is the number of predictions being assessed. We compare RMSEs over three different horizons: the past three years, the past ten years, and the full evaluation sample period. We do this comparison for the full sample which includes the COVID-19 period as well as a sample that ends in 2019:Q4, excluding the effects of the COVID-19 crisis as a robustness check. The results are presented in Table 3 which provides both the raw RMSEs for each model computed using Equation (12) as well as the RMSE for each model relative to the sample mean model. A relative RMSE greater than one implies the model's predictions are less accurate compared to the benchmark model while a relative RMSE less than one implies the model's predictions are more accurate than the benchmark model.^[56]

For the full sample period, model M1 outperforms all other models across each of the three horizons. In relative terms, model M1's RMSEs are over half of those of the sample mean model across the past three-year and ten-year periods and just under three-quarters of the benchmark model for the full sample. The QA model was the only other model that achieved a similar level of performance for the full sample horizon. As previously discussed in relation to Figure 5, this result is primarily because of how well model M1 predicted the significant decline in quarterly GDP growth that occurred in 2020:Q2. This is supported by comparing the model RMSEs in the pre-COVID-19 period. Here, there is no one model that outperforms the others in all periods as was the case when the COVID-19 period was included. Further, all model RMSEs are notably lower and much closer together as well. The models incorporating monthly information are not as dominant either. Indeed, the three ‘M’ models are outperformed by the sample mean model in both the shorter three-year and longer full sample horizons. In contrast, the QA model does narrowly outperform the benchmark model across all three horizons, suggesting that there is always some benefit to using timely information to make predictions. However, it also suggests that there might exist a trade-off between model size and accuracy, especially when making predictions during relatively ‘normal’ periods.

When comparing our results to those of previous studies related to forecasting/nowcasting quarterly GDP growth, it is only fair to focus exclusively on our pre-COVID-19 sample results. In this light, our results still show a clear benefit to using higher frequency (monthly) data for predicting lower frequency (quarterly) data. Both Australian Treasury (2018) and Panagiotelis et al (2019), who each focus on quarterly data, are unable to consistently outperform the sample mean benchmark model. However, Australian Treasury's model can beat the sample mean model once all data on the current quarter are available (the timing of which would be comparable to our M3 and QA models). In contrast, all FA-U-MIDAS models except M3 outperform the sample mean model on average over the last ten years, while the QA version shows outperformance across this timeframe as well as over the last three years and full sample (1988:Q2–2019:Q4).

3.3 Evaluating model performance during the COVID-19 crisis

As shown in Table 3, the three-year horizon which covered the COVID-19 crisis shows substantial divergence in the accuracy of model predictions incorporating monthly and quarterly information. Most of this outcome can be attributed to one time point: June 2020 – the quarter that experienced the brunt of the initial government-mandated COVID-19 lockdowns and the subsequent disruption to economic activity that resulted. The prediction error generated for each model relative to the actual first-release quarterly GDP growth outcome for that period is presented in Figure 6. A value above one means the prediction error was larger than the actual GDP outcome, while a value less than one signifies the prediction error was smaller than the actual GDP outcome.

Note: Errors are relative to first-release quarterly GDP growth in 2020:Q2 (–7 per cent).

Figure 6 helps illustrate how incorporating high frequency (monthly) information greatly improved forecast/nowcast performance for most FA-U-MIDAS model predictions for this quarter, especially model M1 (which includes the month of April 2020 in its nowcast). Model M1 achieved a nowcast error equivalent to roughly one-quarter of the size of the eventual downturn that occurred in quarterly GDP growth (–7 per cent). What is crucial about this from a policymaker's perspective is the nowcast from model M1 was capable of being generated midway through the quarter in question – almost three months before the official figure on GDP would finally be published. Thereby giving policymakers a very timely reading on how the COVID-19 crisis was affecting activity.

Like with the prediction results, the errors for models M2 and M3, which include more timely information, are also both substantially larger than model M1. The performance of the QA model, which uses a temporal aggregated version of the MAI (i.e. three-month average), appears to strike a compromise between the three MIDAS models; suggesting there might be situations when it is beneficial to use temporal aggregated regressors in models, potentially in cases when the model might be otherwise over-parameterised. However, our key result that incorporating timely information can improve model prediction accuracy during downturns corresponds to previous work including Clements and Galvão (2009) (the US recession in 2001), Schorfheide and Song (2015) (the GFC impact on US economic activity in 2008) and Jardet and Meunier (2022) (the COVID-19 pandemic's effect on world GDP growth).

3.4 Assessing the predictive content of the MAI

The relative RMSE results in the previous section indicate that models incorporating monthly information generate more accurate predictions (and smaller errors) compared to the baseline sample mean model. However, to be definitive, it is important we compare model performance using a formal statistical test of equal predictive accuracy.

We cannot use the well-known Diebold-Mariano-West (DMW) t-type test for equal predictive accuracy since we are evaluating nested models (all models include an intercept). Instead, we follow the approaches of Clark and McCracken (2005) and Clements and Galvão (2009) and implement the bootstrap version of the MSE-F test of equal mean squared error (MSE) developed by McCracken (2007).^[57] Let MSE_i denote the MSE from model i for $i\in \left\{\text{AR}\left(1\right)\text{FC,M1,M2,M3,QA}\right\}$ , then the test of equal predictive accuracy of the benchmark sample mean model (i.e. $\mu$ ) and the alternative model specifications considered are implemented using the following test statistic:

where P is the number of predictions being compared. A negative MSE-F implies that model i is less accurate compared to the sample mean model, whereas a positive MSE-F means the model i is more accurate. The bootstrap is used to compute the p-value for the MSE-F test and proceeds as follows. The sample mean model is estimated using the whole sample period of first-release quarterly GDP growth (as recommended by Clements and Galvão (2009)). From the model fit we take the estimated intercept and the variance of the residuals and simulate multiple time series trajectories from the sample mean model assuming Gaussianity.^[58] For each one of the simulated time series trajectories, we apply the same recursive estimation and prediction steps we used with the actual data to calculate the MSE-F statistic for that replication. Note, the MAI is held fixed in each replication. We set the total number of replications in the bootstrap procedure to 1,000. The empirical p-value is calculated as the proportion of MSE-F statistics from the simulations that are larger than the MSE-F statistic computed using actual data. We implement the bootstrapped MSE-F test for the full sample including the COVID-19 period and a shorter sub-sample excluding the COVID-19 period as we did in relation to the RMSE comparisons in Table 3. The results are presented in Table 4.

The MSE-F test results confirm the findings in Table 3 and they also differ depending on whether the test is conducted on the full sample or the pre-COVID-19 sample. For the full sample we strongly reject the null hypothesis of equal predictive accuracy in relation to the sample mean model and all four FA-U-MIDAS models incorporating monthly information. However, the same is not true for the pre-COVID-19 sample, where the null is only rejected for model QA (although model M3 is borderline at the 10 per cent level).^[59]

Overall, these results mirror those of Chauvet and Potter (2013) and Siliverstovs (2020) that relate to the accuracy of model predictions of quarterly GDP growth in the United States changing between expansions and recessions. In our case, the statistical evidence favouring models incorporating more timely information over simpler models based on quarterly information is mostly due to significant outperformance during the three-year period covering the COVID-19 crisis. In contrast, during more ‘normal’ times, the model predictions incorporating monthly information fail to meaningfully improve on those of the benchmark sample mean model. Considering Figure A3, this is not surprising. Australian quarterly GDP growth is serially uncorrelated. However, model QA which includes some information on the current quarter (albeit averaged), was able to consistently outperform the benchmark model in both sample periods. This suggests there might be a trade-off between incorporating more timely information and increasing model complexity.

We have made two important contributions to the factor monitoring and prediction literature related to Australia. First, we developed a monthly activity indicator with a very long history using a ‘supervised’ DFM model, with the explicit goal of providing policymakers with a timely snapshot on prevailing economic conditions. The time span covered by our MAI (45 years) is unmatched by previous work. Second, we have exploited the higher-frequency information imbedded in the MAI in a comprehensive nowcasting exercise covering a 35-year period and show statistically significant outperformance compared to standard benchmark models is possible. In this regard, our work is the first to apply a mixed frequency framework in a systematic manner.^[60] Further, we show that outperformance was greatest during the COVID-19 period, emphasising the benefit of using monthly data.

One curiosity related to our results is unlike many other works in which the prediction of GDP growth becomes more accurate as more data on the quarter comes to hand, our results show the opposite and get less accurate. We speculate this is related to increased parameter uncertainty due to estimating progressively larger models. Further, the COVID-19 crisis caused very large outliers (otherwise known as ‘leverage points’ due to their effect on the estimated regression fit) which can have a substantial effect on parameter estimation.

Despite this, our results do have some encouraging news for policymakers. By using MIDAS-based models incorporating the (timely) MAI, we show it is possible to predict Australian quarterly GDP growth more accurately during crisis periods (such as during the COVID-19 crisis) – a situation when accuracy is needed most. This comes about because the higher-frequency information contained in the MAI means the MIDAS models are quicker to detect abrupt changes, thereby giving policymakers more time to react.

One potential limitation of our work is that we do not redo the pre-selection step to determine the targeted predictor dataset at each time point in the out-of-sample prediction evaluation exercise. Instead, the ranking is done only once and using the full sample. This could bring some issues with our results; however, we are restricted by the unbalanced nature of our dataset which starts with only 17 series and is not as large as in other studies which have also mostly considered balanced datasets. But since we are not seeking to compare predictive accuracy of factor(s) extracted from a targeted predictor dataset to non-targeted predictor datasets this is probably less of a concern. Nonetheless, in future iterations, the targeted predictor dataset should be reviewed and updated as required to ensure it continues to contain only series that are informative about quarterly GDP growth.

In future work we intend to investigate three extensions: non-traditional data; sparsity; and nonlinearity. During the COVID-19 crisis greater use was made of non-traditional data such as mobility. We have not included any of these types of data in our extended dataset, although previous work suggests there might be merit for doing so (e.g. Choi and Varian (2012), who show internet searches can have predictive content). The challenge with some newer non-traditional datasets relates to their relatively short histories, making them harder to incorporate into analysis such as ours. In our work we used a two-step approach to identify the targeted predictor dataset and then extracted the factor afterwards. Recent work from Mosley, Chan and Gibberd (2024) suggests it might be possible to combine these two steps into one by incorporating sparsity using a form of regularisation into the estimation step. Lastly, the DFM is necessarily a linear model. Perhaps there are benefits to nowcasting from considering nonlinear specifications instead.

A.1 Additional MAI dataset details

Note: Dashed line represents the ${\chi }_{\left[3\right]}^2\left(\alpha =0.1\right)$ critical value = 6.25.

Sources: ABS; Ai Group; ANZ; Authors' calculations; Bloomberg; CoreLogic; NAB; RBA; Westpac and Melbourne Institute.

Notes: GDP is first release. Dashed lines represent 95 per cent confidence intervals.

A.2 Additional MAI estimation details

The number of dynamic factors is determined by looking for the second ‘region of stability’ in relation to S_c (i.e. a value of 0) and checking which value of q_c this corresponds to. See Hallin and Liška (2007) for more details.

Sources: Ai Group; Authors' calculations; NAB; RBA; Westpac and Melbourne Institute.

Note: Grey lines represent the underlying ‘targeted predictor’ dataset; blue line represents the MAI.

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