RDP 202103: Financial Conditions and Downside Risk to Economic Activity in Australia Appendix A: Estimating a Financial Conditions Index for Australia
March 2021
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A.1 The financial conditions index dataset
The specific series used to estimate the FCI were chosen to cover a broad range of indicators of financial conditions. The rationale is that financial instability can manifest itself, and impact on the real economy, in a wide variety of ways. This means it is not only important to include variables that have historically shown themselves to be relevant (e.g. housing and commercial property prices) but also to include variables that are emerging as potential sources of future risk (e.g. indicators of financial system complexity). The set of variables used in this paper have considerable, though not complete, overlap with those that have been identified as important in other research (e.g. Aikman et al 2018; Prasad et al 2019).
Our main selection criterion was that the series needed to be available at least since the mid 1990s so as to capture as much time variation as possible, particularly given that historical episodes of financial instability in Australia have so far been few and far between.^{[18]} One downside of this approach is that it excludes some measures that are likely to be very relevant today – and going forward – because they do not have sufficiently long histories. Relatedly, the approach is vulnerable to the criticism that it can overweight measures that happen to be available for longer periods of time.
The resulting set of variables we use can be grouped into the following categories: asset prices (18 series); interest rates and spreads (17); credit and money (14); debt securities outstanding (11); indicators of leverage (5); indicators of banking sector risk (4); indicators of financial system ‘complexity’ (2); indicators of financial market risk (2); and surveys' indicators of businesses' and consumers' views on financial conditions (2). Our dataset spans the sample period 1976:Q3 to 2020:Q3, however, many financial variables we use only have a relatively short history which means our dataset is unbalanced. It starts with 37 series, increases to 54 series by 1987 and to the full dataset of 75 series by mid 1994 but, due to data availability, there are only 42 series for September quarter 2020 (Figure A1).
Before estimating the FCI we transform all variables in our dataset to be stationary and standardise them to have zero mean and unit variance.^{[19]} Because all of our variables are nominal, we control for potential structural breaks in the mean of each series in our dataset following the introduction of inflation targeting by the RBA in 1993 by ‘dynamically demeaning’ each series using a rolling 10year backwardlooking estimate of the sample mean (rather than using the full sample mean). Using a rolling estimate of the sample mean allows us to be agnostic about the precise date the break occurred and is preferable to estimating a specific break date for each series in the dataset, which could be subject to large estimation uncertainty.^{[20]} The choice of a 10year window follows Kamber, Morley and Wong (2018), but is not specifically related to the length of business or financial cycles in our case. Note, dynamically demeaning all series even if no series has a structural break in the mean should not cause any estimation problems. If there is no structural break in a particular series then the rolling window estimated sample mean for that series should be equal to its full sample mean.
Some previous papers have ‘purged’ the financial data of correlations with measures of economic activity before estimating the FCI by using the residuals from regressions of the financial variables on a set of variables related to ‘macroeconomic conditions’ (e.g. Hatzius et al 2010). However, we follow Kapetanios et al (2017) who argue that purging the data with a prior regression may reduce its usefulness as a summary measure. For example, if the only common shock in the economy at a given point in time was financial in nature, then this type of analysis would amount to purging the financial series of precisely the information we are aiming to summarise. Table A1 provides a complete list of all series we use to estimate the FCI and specifies the source, the start and end dates as well as the specific transformation we apply to each series.
No  Variable  Source  Economy  Start date  End date  Transformation code 

Survey measures  
1  Business: difficulty getting finance  ACCIWBC  Aus  1966:Q2  2020:Q3  FD 
2  Consumer: family finances now  WBCMI  Aus  1974:Q4  2020:Q3  LV 
Interest rates and spreads  
3  Overnight cash rate (OCR)  RBA  Aus  1976:Q3  2020:Q3  FD 
4  3month bank bill rate  ASX; RBA  Aus  1976:Q3  2020:Q3  FD 
5  3year Australian Government security (AGS) yield  RBA  Aus  1992:Q3  2020:Q3  FD 
6  5year AGS yield  RBA  Aus  1976:Q3  2020:Q3  FD 
7  10year AGS yield  RBA  Aus  1969:Q3  2020:Q3  FD 
8  Spread: 3month bank bill to OCR  ASX; RBA  Aus  1972:Q1  2020:Q3  LV 
9  Spread: 3year AGS to OCR  RBA  Aus  1992:Q3  2020:Q3  LV 
10  Spread: 5year AGS to OCR  RBA  Aus  1969:Q3  2020:Q3  LV 
11  Spread: 10year AGS to OCR  RBA  Aus  1976:Q3  2020:Q3  LV 
12  Federal funds rate (FFR)  FRED  US  1959:Q3  2020:Q3  FD 
13  3month Treasury bill (Tbill) yield  FRED  US  1959:Q3  2020:Q3  FD 
14  3year Treasury bond (TB) yield  FRED  US  1959:Q3  2020:Q3  FD 
15  10year TB yield  FRED  US  1959:Q3  2020:Q3  FD 
16  Spread: 3month Tbill to FFR  FRED  US  1959:Q3  2020:Q3  LV 
17  Spread: 3year TB to FFR  FRED  US  1959:Q3  2020:Q3  LV 
18  Spread: 10year TB to FFR  FRED  US  1959:Q3  2020:Q3  LV 
19  Spread: 10year AGS to 10year USTB  RBA; FRED  Aus/US  1969:Q3  2020:Q3  LV 
Credit and money  
20  Total credit  RBA  Aus  1976:Q3  2020:Q3  LD 
21  Housing credit  RBA  Aus  1976:Q3  2020:Q3  LD 
22  Personal credit  RBA  Aus  1976:Q3  2020:Q3  LD 
23  Business credit  RBA  Aus  1976:Q3  2020:Q3  LD 
24  Owneroccupier housing loan approvals (excl refinancing)  ABS  Aus  1985:Q1  2020:Q2  LD 
25  Investor housing loan approvals (excl refinancing)  ABS  Aus  1985:Q1  2020:Q2  LD 
26  Commercial fixed term loan approvals (excl refinancing)  ABS  Aus  1985:Q1  2020:Q2  LD 
27  Commercial revolving credit approvals (excl refinancing)  ABS  Aus  1985:Q1  2020:Q2  LD 
28  Personal fixed term loan approvals (excl refinancing)  ABS  Aus  1985:Q1  2020:Q2  LD 
29  Personal revolving credit approvals (excl refinancing)  ABS  Aus  1985:Q1  2020:Q2  LD 
30  M1  RBA  Aus  1975:Q1  2020:Q3  LD 
31  M3  RBA  Aus  1976:Q3  2020:Q3  LD 
32  Broad money  RBA  Aus  1976:Q3  2020:Q3  LD 
33  Money base  RBA  Aus  1975:Q1  2020:Q3  LD 
Asset prices  
34  Dwelling price index  CoreLogic  Aus  1980:Q1  2020:Q3  LD 
35  House price index  CoreLogic  Aus  1980:Q1  2020:Q3  LD 
36  Apartment price index  CoreLogic  Aus  1980:Q1  2020:Q3  LD 
37  Dwelling price index  FRED  US  1987:Q1  2020:Q2  LD 
38  All commercial property return index  MSCI  Aus  1984:Q4  2020:Q2  LD 
39  Retail property return index  MSCI  Aus  1984:Q4  2020:Q2  LD 
40  Office property return index  MSCI  Aus  1984:Q4  2020:Q2  LD 
41  Industrial property return index  MSCI  Aus  1984:Q4  2020:Q2  LD 
42  ASX 200 Index  Refinitiv  Aus  1973:Q1  2020:Q3  LD 
43  ASX 200 Financials Index  Refinitiv  Aus  1973:Q1  2020:Q3  LD 
44  ASX 200 Real Estate Index  Refinitiv  Aus  1973:Q1  2020:Q3  LD 
45  ASX 200 Resources Index  Refinitiv  Aus  1973:Q1  2020:Q3  LD 
46  ASX 200 Industrials Index  Refinitiv  Aus  1973:Q1  2020:Q3  LD 
47  S&P 500 Index  FRED  US  1960:Q1  2020:Q3  LD 
48  RBA Index of Commodity Prices (AUD)  RBA  Aus  1959:Q3  2020:Q3  LD 
49  Gold (3pm London bullion market, USD)  FRED  US  1968:Q2  2020:Q3  LD 
50  Crude oil (West Texas intermediate, USD)  FRED  US  1968:Q2  2020:Q3  LD 
51  Australian dollar tradeweighted index  RBA  Aus  1970:Q3  2020:Q3  LD 
Debt securities outstanding  
52  Shortterm: Australia: banks  APRA; Austraclear Limited; RBA  Aus  1992:Q4  2020:Q2  LD 
53  Shortterm: Australia: nonfinancial corporations  Austraclear Limited; RBA  Aus  1992:Q4  2020:Q2  LD 
54  Longterm: banks  Bloomberg; RBA  Aus  1992:Q4  2020:Q2  LD 
55  Longterm: Australia: nonfinancial corporations  Bloomberg; RBA  Aus  1992:Q4  2020:Q2  LD 
56  Shortterm: Australia: government  AOFM; Austraclear Limited; RBA  Aus  1992:Q4  2020:Q2  LD 
57  Longterm: Australia: government  AOFM; RBA; State borrowing authorities  Aus  1992:Q4  2020:Q2  LD 
58  Longterm: overseas: nongovernment  ABS; Bloomberg; Private Placement Monitor; RBA  Aus  1992:Q4  2020:Q2  LD 
59  Shortterm: Australia: assetbacked securities  ABS; Bloomberg; RBA  Aus  1992:Q4  2020:Q2  LD 
60  Longterm: Australia: assetbacked securities  ABS; Bloomberg; RBA  Aus  1992:Q4  2020:Q2  LD 
61  Longterm: overseas: assetbacked securities  ABS; Bloomberg; RBA  Aus  1992:Q4  2020:Q2  LD 
62  Residential mortgagebacked securities  ABS; Bloomberg; RBA  Aus  1992:Q4  2020:Q2  LD 
Banking sector  
63  Tier 1 capital ratio  APRA; RBA  Aus  1989:Q2  2020:Q2  FD 
64  Nonperforming assets ratio  APRA; RBA  Aus  1990:Q2  2020:Q2  FD 
65  Distance to default  RBA  Aus  1983:Q1  2020:Q2  LV 
66  Wholesale debt spread to AGS  Bloomberg; RBA  Aus  1994:Q2  2020:Q2  FD 
Financial system complexity (ratio)  
67  Total financial institutions' assets to nominal GDP  ABS; RBA  Aus  1990:Q1  2020:Q2  LD 
68  Total offbalance sheet business to total fixed income assets  ABS; RBA  Aus  1990:Q1  2020:Q2  LD 
Leverage measures (ratio)  
69  Household debt to assets  ABS; RBA  Aus  1988:Q3  2020:Q2  LD 
70  Household debt to income  ABS; RBA  Aus  1988:Q2  2020:Q2  LD 
71  Household interest payments to income  RBA  Aus  1977:Q1  2020:Q2  LD 
72  Current account balance to nominal GDP  ABS  Aus  1959:Q3  2020:Q2  FD 
73  Net total foreign liabilities to nominal GDP  ABS  Aus  1988:Q3  2020:Q2  FD 
Risk indicators  
74  Chicago Board Options Exchange equity volatility  FRED  US  1986:Q4  2020:Q3  LV 
75  Moody's corporate bond yield spread: BAA to AAA  FRED  US  1959:Q3  2020:Q3  LV 
Notes: ‘ABS’ is Australian Bureau of Statistics, ‘ACCIWBC’ is Australian Chamber of Commerce and Industry–Westpac, ‘AOFM’ is the Australian Office of Financial Management, ‘FRED’ is Federal Reserve Economic Database, Federal Reserve Bank of St. Louis, ‘WBCMI’ is Westpac and Melbourne Institute; ‘Transformation code’ indicates the method used to transform the data to be stationary if necessary, ‘FD’ indicates first difference, ‘LD’ indicates log difference and ‘LV’ indicates level 
A.2 Constructing the financial conditions index
The general form of the DFM we use to estimate the FCI follows Bai and Wang (2015) and is defined as:
where y_{t} is N × 1 vector of observables, f_{t} is a q × 1 vector of the dynamic factors, and ${\text{\Lambda}}_{j}$ is the dynamic factor loadings for f_{t – j} with j = 0,1,...,s and t = 1,...,T. The dynamic factors follow a VAR(p) process with ${\text{\Phi}}_{j}$ 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}) which is irrespective of s and p.
The covariance matrix of ${\epsilon}_{t}$ in the measurement equation 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 $\text{\mathbb{E}}\left[{\epsilon}_{t}{\eta}_{t}^{\xb4}\right]=0$ (i.e. the 2 noise processes are independent). This specification of the DFM has 2 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 is assumed to be captured by a VAR(p) process. 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.^{[21]} The statespace representation of the DFM specified by Equation (A2) is:
The measurement equation takes the form of a static factor model (Stock and Watson 2002) with r = q (s + 1) static factors. Now, define k = max(p,s + 1) and set ${\text{\Phi}}_{p+1}=\cdot \cdot \cdot ={\text{\Phi}}_{k}$ if k = s + 1 > p.
Let ${F}_{t}={\left[{f}_{t}^{\xb4},\mathrm{...},{f}_{tk+1}^{\xb4}\right]}^{\xb4}$, then the N × qk factor loadings matrix $\text{\Lambda}$ the qk × qk matrix of VAR coefficients $\text{\Phi}$ and the qk × q selector matrix G have the following form:
when k = p > s, let ${F}_{t}^{*}=\left[{f}_{t}^{\xb4},\mathrm{...}{f}_{ts}^{\xb4}\right]$ be a subvector of F_{t}. Then the factor loadings matrix $\text{\Lambda}$ and the VAR coefficient matrix $\text{\Phi}$ become:
Note, the selector matrix G does not change in this alternative setting. The benefit of this modelling strategy over others, such as traditional static factor models based on principal components analysis (PCA), is that it explicitly incorporates dynamics into the estimation and can easily and efficiently handle unbalanced datasets such as the one we use.
When implementing the DFM we assume one common factor (i.e. q = 1) is adequate to capture the common variation we are most interested in from our dataset. This decision was made because we want a single index to represent ‘financial conditions’. Multiple factors would require us to combine them into a single index and it is unclear what the optimal way to do this is. Moreover, the modified Bai and Ng (2002) information criterion proposed by Coroneo, Giannone and Modugno (2016) suggests one factor is sufficient.^{[22]}
The dynamics of the factor are assumed to follow an AR(1) process (i.e. p = 1). This was determined by the Schwarz information criterion (SIC), together with plots of the sample autocorrelation function (ACF) and sample partial ACF for the common factor based on an initial estimate of the common factor obtained by PCA.
We choose to include one lag of the factor in the measurement equation (i.e. s = 1) because some of the series in our dataset are market determined. These series will tend to be less persistent (and potentially more volatile) than the other series in our dataset which are not market determined. A DFM with dynamic loadings will be able to accommodate this explicitly.
Following Doz, Giannone and Reichlin (2011, 2012) we estimate the DFM by quasimaximum likelihood.^{[23]} Estimation consists of 2 parts. First, the parameters of the statespace representation of the DFM are obtained by applying PCA to a balanced subset of the dataset which contains all 75 series. Second, the Kalman filter is applied on the original unbalanced dataset in order to obtain estimates of the factors using only the information available at each point in time.
For the observations Y = {y_{1},...,y_{T}}, factors F = {F_{1},...,F_{T}} and parameter vector $\theta =\left\{\text{\Lambda ,\Phi},R,Q\right\}$ the complete data log likelihood is given by:
We estimate the parameter vector $\theta $ using the expectationmaximisation (EM) algorithm and the Kalman filter and RauchTungStriebel (RTS) smoother recursions (for more details, see Anderson and Moore (1979)). To do this, we set the initial values in the statespace equations using the first r principal componentsbased estimates of the factors and OLS estimates of the parameters $\text{\Lambda ,\Phi},R$ and Q treating the principal components factors as the true common factors. From these starting values we then alternate between 2 steps. First, we estimate the expected value of the common factors given the data and previously estimated model parameters (the ‘Estep’). This involves running the Kalman filter and RTS smoother recursions and getting a new estimate of the common factors denoted by $\mathbb{E}\left[{F}_{t}Y;\widehat{\theta}\right]$ where $\widehat{\theta}$ is the vector of estimated model parameters from the previous step. Then using these updated estimates of the common factors we compute new parameter estimates by maximising the expected log likelihood with respect to $\theta $ (the ‘Mstep’).
To account for missing observations in y_{t} we define the known matrix W_{t} which is a diagonal matrix of size N with the ith diagonal element equal to zero if y_{i,t} is missing and equal to one otherwise. The matrix W_{t} acts a selector matrix and ensures only the available data are used in the Kalman filter and RTS smoother recursions. The updating equations for the parameters $\text{\Lambda ,\Phi ,}R$ and Q are as follows:
To compute the next $\text{\Lambda}$ we use:
To compute the next $\text{\Phi}$ we use:
To compute the next R we use:
To compute the next Q we use:
The initial state F_{0} and initial state variance P_{0} are given by:
See Ghahramani and Hinton (1996) and Bańbura and Modugno (2014) for more details on these calculations. We judge convergence to be when c_{m} is less than 10^{–4}, with c_{m} given by:
and m = 1,...,M is the number of evaluations needed to achieve convergence up to a maximum M set by the researcher. We set M = 500 but we only needed 8 evaluations in our case.
A convenient feature of this specification is that the computational complexity of the Kalman filter and RTS smoother recursions depends only on the size of the state. In our case this corresponds to r and is independent of the size of the crosssection N. Note, while the EM algorithm will converge to a maximum, it is not guaranteed to find the global maximum and can converge to a local maximum. We can reduce the probability of this occurring by starting the algorithm with the PCbased estimates which are consistent for large crosssections (see Stock and Watson (2002) and Bai and Ng (2002)).
A common issue with estimating factor models is the lack of identification. In our case this means the likelihood of our model is invariant to any invertible linear transformation of the factors. That is, for any invertible matrix H the parameters $\theta $ = $\left\{\text{\Lambda ,\Phi ,}R,Q\right\}$ and ${\theta}_{H}=\left\{\text{\Lambda}{H}^{1},H\text{\Phi}{H}^{1},R,HQ{H}^{\xb4}\right\}$ are observationally equivalent and hence $\theta $ is not identifiable from the data. In order to achieve identifiability of $\theta $ , we need to impose an identifying restriction. We implement the restriction ‘DFM2’ suggested by Bai and Wang (2015).^{[24]} This identifying restriction specifies the first q × q block of ${\text{\Lambda}}_{0}$ in $\text{\Lambda}$ to be an identity matrix:
where ${\text{\Lambda}}_{0}^{*}$ is an (N – q) × q unrestricted block of ${\text{\Lambda}}_{0}$. This parameterisation of ${\text{\Lambda}}_{0}$ is also called the ‘named factor’ normalisation because it associates each factor with a specific variable (Stock and Watson 2016).
When implementing this identifying restriction it is important to order the variables in Y so that the first q variables are the naming variables. In practice, the naming variables need to be reasonably different from each other and sufficiently representative of groups of other variables so that the innovations to their common components span the same space as the factor innovations. In our work we choose to put the series ‘survey: difficulty getting finance for business’ as the first series. Increases in this series reflect greater difficulty for businesses getting access to finance while decreases in this series reflect less difficulty for businesses in getting access to finance. A benefit of this normalisation is that it also identifies the sign of the estimated common factor. It is common for more restrictive financial conditions to be reflected by increases in the FCI and for more expansionary financial conditions to be reflected by decreases in the FCI (see Brave and Butters (2010)). Hence, being able to impose this sign convention with our FCI is beneficial. The named factor normalisation means the interpretation of the factor loadings on the other variables changes. They become measures of the correlation of each series with the factor relative to the named factor series.
To impose this identifying linear restriction in estimation we need to modify the Mstep used to estimate $\text{\Lambda}$ to incorporate the linear restriction specified by:
where ${M}_{\text{\Lambda}}$ is a selector matrix with dimension q^{2} × (N × r) with 0 or 1 elements specifying the coefficients that will be restricted. The vector ${h}_{\text{\Lambda}}$ is of dimension q^{2} × 1 also with 0 or 1 elements and are the values that the coefficients will be restricted to be. The updating equation incorporating the identifying restriction for $\text{\Lambda}$ becomes:
where ${\text{\Lambda}}^{r}$ is the estimated factor loadings with the identifying restriction and ${\text{\Lambda}}^{u}$ is the unrestricted estimated factor loadings given previously by Equation (A6), see Wu, Pai and Hosking (1996) and BaĆbura and Modugno (2014) for more information.
A.3 Which series are most important for financial conditions?
The sign and magnitude of the combined values of the 2 factor loadings for each series indicate the relative strength and direction of that series' relationship with the estimated FCI. Figure A2 highlights the 10 most important series by positive and negative combined weights. Series with a negative (positive) combined weight tend to move in the opposite (same) direction to the FCI, falling (rising) when the FCI indicates financial conditions are restrictive and rising (falling) when the FCI indicates financial conditions are expansionary.
The series which are most strongly related to the contemporaneous value of the FCI tend to have negative weights. These same series also tend to be negatively related to the oneperiod lag of the FCI as well. These include property prices, growth in credit aggregates and one survey measure related to consumer expectations of their finances. Of these, commercial property has the highest combined weighting and as a result has the strongest relationship with the FCI. This is unsurprising given the highly procyclical nature of commercial property returns during the only 2 episodes of significant financial instability in our sample. Coupled with the fact that the commercial property sector has historically been a relatively important source of risk to banks' balance sheets (Ellis and Naughtin 2010), these results underscore the importance of monitoring developments in this sector on an ongoing basis.
In contrast, only onequarter of the dataset are positively related to the FCI, with the magnitude of the combined weights typically much smaller than those series with negative signs. The strongest positive correlations belong mostly to interest rate spreads. Note, we have not included the series ‘survey: difficulty getting finance for business’ since the first factor loading is restricted to be unity by design (see Appendix A.2).
A.4 Robustness checks for the FCI
We undertake 2 checks of our preferred estimated FCI to confirm its soundness as a measure of financial conditions. The first is a comparison of the DFM methodology against an alternative method for estimating the FCI. The second is a comparison of our preferred estimated FCI (estimated using the DFM approach) and the named factor series ‘survey: difficulty getting finance for business’. The named factor series is the one we choose to restrict the factor loading to be unity in estimation for identification purposes. Both comparisons are shown in Figure A3.
The top panel compares our preferred estimated FCI with another FCI produced using a static factor model (SFM). The DFM technique is preferable to the SFM technique since it allows the FCI to be estimated over a longer time period (SFMs are estimated by PCA and require balanced datasets). Further, since the DFM explicitly incorporates dynamics, it produces an indicator that appears smoother than the indicator based on the SFM estimate.
The bottom panel shows the estimated FCI and the named factor series ‘survey: difficulty getting finance for business’. Both series display a reasonably similar ‘low frequency’ pattern, but the movements between the two series are not exactly the same. This shows that the named factor series is not significantly influencing the profile of the estimated FCI besides helping to identify its scale, as intended.
A further robustness check of the DFM methodology is to formally test whether the dataset as a whole supports the inclusion of a lag of the factor in the measurement equation. We test this using a likelihood ratio ( LR ) test.^{[25]} This is easy to do since we get the log likelihood as a byproduct of the Kalman filter. To compute the LR statistic we estimate the DFM twice. Once with the lag included to get the value of the unrestricted log likelihood ${\U0001d4db}_{u}$ and again with the lag excluded to get the value of the restricted log likelihood ${\U0001d4db}_{r}$ The LR test statistic is then computed as:
where k is the number of restrictions to be tested. Under the null hypothesis the test is distributed as chisquared with k degrees of freedom. The test result is shown in Table A2. The test statistic is very large at 1,091 and with k = 75 parameter restrictions the associated pvalue is effectively zero.
This suggests there is little evidence against the DFM methodology we have used to construct the FCI.
Number of parameter restrictions ( k )  75 

Unrestricted log likelihood ( ${\U0001d4db}_{u}$ )  −16,145.5 
Restricted log likelihood ( ${\U0001d4db}_{r}$ )  −16,691.2 
Test statistic  1,091.4 
pvalue  0.0 
Note: Test statistic is computed as −2$\left({\U0001d4db}_{r}{\U0001d4db}_{u}\right)$ which is distributed as chisquared with k degrees of freedom under the null 
A.5 Assessing the relationship between financial conditions and economic outcomes
We start by assessing the FCI's ability to predict 5 separate summary measures of economic activity: real GDP, real household final consumption expenditure, the unemployment rate, employment and real nonmining business investment. We examine both quarterly and yearended changes in these variables over the period 1976:Q4 to 2020:Q2.^{[26]}
Figure A4 plots the FCI against the 5 measures of economic activity for quarterly and yearended time horizons. Each series has been standardised to have zero mean and unit variance. Most series exhibit reasonable contemporaneous relationships with the FCI, although the strength varies. Four of the measures of economic activity are negatively related to financial conditions (more restrictive financial conditions correspond to weaker growth), while change in unemployment rate has a positive relationship (more restrictive financial conditions are associated with an increase in the unemployment rate).
Estimates of the sample crosscorrelation between the measures of activity and the FCI for both time horizons are provided in Table A3.
The contemporaneous sample correlations between the measures of economic activity and the FCI are strongest for the 2 labour market variables and weakest for consumption. Across variables, correlations tend to be stronger for yearended, rather than quarterly, changes reflecting the fact that yearended changes are relatively smoother. The sample crosscorrelation between these measures of economic activity and the FCI at different lag lengths suggest there might be a lead/lag relationship between the FCI and the various measures of economic activity.
Quarterly  Yearended  

Contemporaneous  Maximum^{(a)}  Lag^{(b)}  Contemporaneous  Maximum^{(a)}  Lag^{(b)}  
GDP  −0.27  −0.27  0  −0.43  −0.45  1  
Consumption  −0.21  −0.22  1  −0.30  −0.35  1  
Unemployment rate  0.44  0.44  0  0.51  0.55  1  
Employment  −0.46  −0.47  1  −0.60  −0.67  1  
Business investment  −0.28  −0.28  −1  −0.46  −0.46  0  
Notes: All series are in growth rates besides the unemployment rate which is in differences

However, sample crosscorrelations between two serially correlated time series can sometimes indicate spurious relationships (see Cryer and Chan (2008)). To formally assess these potential lead/lag relationships we use Granger causality tests. This attempts to determine if the lags of one time series have information that is useful for predicting another time series over and above the information already contained in the lags of that second series. The test can be sensitive to the choice of lag length, so we test for lag lengths from one to four quarters.^{[27]} We can infer Granger causality when there is evidence that the FCI Grangercauses the measure of economic activity and no evidence that the measure of economic activity Grangercauses the FCI. The results are in Table A4.
The results suggest that the FCI does contain predictive information about the 5 measures of economic activity at a quarterly time horizon, although the results for household consumption are only significant at the 10 per cent level for the first lag. At a yearended time horizon the results are similar for GDP, household consumption, the unemployment rate and employment. In contrast, the FCI only provides additional predictive information for business investment at longer lag lengths (i.e. lags greater than 2).
Lag length  

1  2  3  4  
Quarterly  
GDP does not Grangercause FCI  0.85  1.00  1.00  1.00 
FCI does not Grangercause GDP  0.02  0.00  0.00  0.00 
HFCE does not Grangercause FCI  1.00  1.00  1.00  1.00 
FCI does not Grangercause HFCE  0.09  0.00  0.01  0.01 
UR does not Grangercause FCI  1.00  1.00  1.00  1.00 
FCI does not Grangercause UR  0.00  0.00  0.00  0.00 
EMP does not Grangercause FCI  0.74  0.23  0.26  0.74 
FCI does not Grangercause EMP  0.00  0.00  0.00  0.00 
NMBI does not Grangercause FCI  0.78  0.92  0.72  0.85 
FCI does not Grangercause NMBI  0.00  0.02  0.00  0.00 
Yearended  
GDP does not Grangercause FCI  1.00  1.00  1.00  1.00 
FCI does not Grangercause GDP  0.02  0.00  0.00  0.00 
HFCE does not Grangercause FCI  0.43  0.73  0.28  0.59 
FCI does not Grangercause HFCE  0.06  0.00  0.00  0.03 
UR does not Grangercause FCI  0.27  1.00  1.00  1.00 
FCI does not Grangercause UR  0.00  0.00  0.00  0.00 
EMP does not Grangercause FCI  0.61  0.90  1.00  1.00 
FCI does not Grangercause EMP  0.00  0.00  0.00  0.00 
NMBI does not Grangercause FCI  1.00  1.00  1.00  1.00 
FCI does not Grangercause NMBI  0.04  0.11  0.00  0.00 
Notes: All pvalues have been adjusted using the Bonferroni correction method; ‘GDP’ is the growth rate of real gross domestic product; ‘HFCE’ is the growth rate of real household final consumption expenditure; ‘UR’ is the change in the total unemployment rate; ‘EMP’ is the growth rate of total employment; ‘NMBI’ is the growth rate of real business investment excluding mining investment 
Footnotes
This decision also has a practical purpose. We initialise our DFM using parameter estimates derived from principal components analysis which requires a balanced dataset. Hence, a dataset with lots of very short series could result in less precise starting values which will affect the estimation of the FCI. [18]
Variables are made stationary by taking logs and/or differences as appropriate. [19]
This will also be useful to correct for any structural drift in the mean of any series as well. [20]
Bai and Wang regard the specification to be a static factor model when there are no lags in the measurement equation (i.e. s = 0). [21]
Note, this information criterion is only designed to be used to determine the number of static factors in a dataset and not the number of dynamic factors in a dataset. Our dynamic factor model with one factor and one lag can be rewritten as a static factor model with 2 factors. As a robustness check we test whether the inclusion of the lag (i.e. the second static factor) is supported by the data and we find that it is. These results are in Appendix A.4. [22]
Estimation is ‘quasi’ maximum likelihood because the model is misspecified. This comes from assuming that R the covariance matrix of the idiosyncratic component is diagonal. Additionally, we also assume that both noise processes are Gaussian. However, in large samples, this has been shown to be no issue (see Doz et al (2012), Bai and Li (2016) and Barigozzi and Luciani (2019)). [23]
The identifying restriction ‘DFM2’ is similar to the identifying restrictions ‘PC3’ suggested by Bai and Ng (2013) and ‘IC1’ suggested by Bai and Li (2016). [24]
Our decision to use the LR test relies on Bai and Wang (2015)'s suggestion that a Wald test for parameter restrictions in the DFM is feasible and a Wald test is asymptotically equivalent to a LR test (see Buse (1982)). [25]
The original frequency of the unemployment rate and the employment series is monthly. We convert both series to a quarterly frequency by taking the quarterended monthly value. [26]
We chose to use a maximum of 4 lags based on sample crosscorrelation analysis in Table A2. [27]