Technical appendix to Vacancies, unemployment and labour market slack in New Zealand

September 2020 AN2020/7

 Technical appendix to
 Vacancies, unemployment and
 labour market slack in New
 Zealand
 Finn Robinson

 Reserve Bank of New Zealand Analytical Note Series

 ISSN 2230-5505 Reserve Bank of New Zealand
 PO Box 2498
 The Analytical Note series encompasses a range of types of background Wellington
 papers prepared by Reserve Bank staff. Unless otherwise stated, views NEW ZEALAND
 expressed are those of the authors, and do not necessarily represent the
 views of the Reserve Bank. www.rbnz.govt.nz

Appendix A: Vacancy data sources and issues
A key limitation with empirical studies of the Beveridge curve and matching function is the lack of quality
vacancy data. Often the data are only available over a short time period and vacancies may not be measured
consistently across countries (Consolo and da Silva, 2019).

In New Zealand, two important data sources for vacancies data have been the ANZ Bank job advertisement
series and the Ministry of Business, Innovation, and Employment (MBIE)’s Jobs Online data. The ANZ series
stretches from 1990 to 2018 and counts advertisements placed in newspapers and online. See Silverstone
(2004) for an early example of labour market analysis carried out using this dataset. The ANZ series was
discontinued in 2018 (figure A.1).

The MBIE Jobs Online data is an index which tracks changes in the number of vacancies advertised online. The
data are available at occupational, industry, and regional levels. The vacancy data is reported as an index for
‘commercial sensitivity reasons’ (Fale and Tuya, 2010, p. 3). The data is sourced from four online jobs
platforms: SEEK, TradeMe Jobs, Education Gazette, and Kiwi Health jobs. The Jobs Online data are available
from 2010 and continue to be published. Currently, the base year for the MBIE Jobs Online index is 2010. Since
this Note focuses on aggregate variables, the All Vacancies Index is used, which is based on all of the job-ads
captured in Jobs Online (MBIE, 2018). This index is plotted in Figure A.1.

One concern with counting vacancies posted on multiple job platforms is the possibility of duplicates. For
example, if a job is posted on both SEEK and TradeMe Jobs then it could be counted twice even though there is
only one vacant position. When MBIE receive raw data from its suppliers a few days after the reference month,
part of their data-processing involves removing such duplicates. They check both for duplicates on the same
website and across each of their source websites (MBIE, 2018). The ANZ data were not cleaned in this way, so
the level of vacancies may be overstated by this measure.

 Figure A.1: ANZ online job advertisements and MBIE All Vacancies Index

Source: ANZ, MBIE, Hall and McDermott (2016), author estimates.
Note: Grey shading indicates the 2008 recession in New Zealand, based on the Hall and McDermott (2016) dates.

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In order to construct a long time series of vacancy data in New Zealand, it is necessary to combine the ANZ and
MBIE job vacancies data. Since the MBIE All Vacancies Index captures online advertisements, only the ANZ
count of online vacancies is used, starting in 2000 Q1.

Following the method used by Dutu, Holmes, and Silverstone (2016) the All Vacancies Index is regressed
against the ANZ advertisements series over the period for which they are both available (2010 Q4 – 2018 Q4).
Then using the estimated coefficients from that regression, quoting Dutu, Holmes and Silverstone I ‘backward
forecast’ from 2009 Q4 to 2000 Q1 to obtain a proxy for the MBIE vacancies data over this period (p. 92). Figure
A.2 plots the MBIE All Vacancies Index in red, with the back-cast constructed from the ANZ advertisements
data in blue.

 Figure A.2: MBIE All Vacancies Index and ANZ-based proxy

 Source: ANZ, MBIE, Hall and McDermott (2016), author estimates.
 Note: Grey shading indicates the 2008 recession in New Zealand, based on the Hall and McDermott (2016)
 dates.

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With this splice in place, we now have a time series of vacancies spanning from 2000 Q1 to 2019 Q4 – 80 data
points. While this is by no means a large sample, it is an improvement on the short All Vacancies Index which
only has 37 data points. Figure A.3 plots the spliced online vacancy index.
 Figure A.3: Spliced online vacancy index

 Source: ANZ, MBIE, Hall and McDermott (2016), author estimates.
 Note: Vertical line indicates the start date of the All Vacancies Index. Data points prior to the vertical line are
 estimated using the ANZ job advertisements data. Grey shading indicates the 2008 recession in New Zealand,
 based on the Hall and McDermott (2016) dates.

Appendix B: Econometric approach
B.1: Econometric model
The aggregate matching function summarises the complicated process by which jobs are created in a frictional
labour market, where unemployed people search for jobs and firms post vacancies to try and fill empty positions
(Blanchard and Diamond, 1989a). The matching function can be written as;
 
 = (1)

where hires ( ) is the number of jobs created each quarter, vacancies ( ) is the number of vacancies posted
by firms, and job-seekers ( ) is the number of people looking for a job. is interpreted as matching efficiency
and is akin to the Solow residual in a Cobb-Douglas production function (Borowczyk-Martins, Jolivet, and
Postel-Vinay, 2013).

When = 1 − β this means the matching function has constant returns to scale (CRS) (Borowczyk-Martins,
Jolivet, and Postel-Vinay, 2013). Under CRS, if unemployment and vacancies were to double, the number of
jobs created would also double.

Matching functions have been estimated for many countries using many statistical methods and different data
(see Petrongolo and Pissarides (2001) for an overview). The estimation is often carried out in log-levels. For
example, Blanchard and Diamond (1989a) estimate the following regression:

 ln = 0 + 1 ln + 2 ln + 3 (2)

where is a time trend, which is often found to be negative in these regressions. A negative time trend implies
that matching efficiency has declined over the sample period (Petrongolo and Pissarides, 2001).

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For the current Note, an autoregressive distributed lag model (ARDL) approach is chosen to estimate equation
2. This approach has the advantage that, when properly specified, ordinary least squares (OLS) estimates of
the parameters of the model will be consistent, and valid standard errors will be obtained for inference, even
when one or more of the variables in the regression are integrated of order one (Pesaran and Shin, 1999).

In addition, the ARDL model can be transformed into an error correction representation. This allows us to
determine whether there is a long-run equilibrium relationship between job-creation, job-seeking, and vacancies
whilst controlling for short-run dynamics (Pesaran and Shin, 1999). The ARDL approach is used by Raines and
Baek (2016) to estimate the Beveridge curve in the US, allowing them to estimate the equilibrium relationship
between vacancies and unemployment whilst controlling for the short run dynamics of these variables.

Bell (1997) conducts a cointegration analysis of matching functions in France, Great Britain and Spain using the
ARDL approach, and finds evidence of the existence of an aggregate matching function in each country.
However, Bell does not find evidence in favour of CRS, which highlights the importance of testing this
assumption before imposing it on the data.

The ARDL model to be estimated is specified as follows:
 
 ( ) = 0 + 1 + ∑ ( − ) + ∑ ( − ) + ∑ ( − ) + (3)
 =1 =0 =0

where the parameters , , are the number of lags of , , and included in the regression.1 is a trend
which should capture any trend deterioration in labour market performance over the sample.

We can re-parameterise the ARDL model as an error correction model:

 ∆ ( ) = 0 + + ∑ ∆ ( − ) +
 =1
 
 ∑ ∆ ( − ) + ∑ ∆ ( − ) +
 =0 =0
 1 ( −1 ) + 1 ( −1 ) + 1 ( −1 ) + 

As Raines and Baek (2016) note in their cointegration analysis of the Beveridge curve, we can now test the
existence of a long-run relationship. The null hypothesis of no cointegration is : 1 = 1 = 1 = 0 against the
alternative : 1 ≠ 0, 1 ≠ 0, 1 ≠ 0. The null hypothesis is assessed using the bounds testing approach
developed by Pesaran, Shin, and Smith (2001). The results of the test for cointegration are reported in table
B.4.2.

B.2: Data selection
Petrongolo and Pissarides (2001) offer a survey of the early literature on estimating matching functions. There
is considerable variation in the choice of data used to measure the dependent and independent variables in the
matching function, with the choice of data often restricted by availability.

A common measure of job creation or hires ( ) is the number of people moving from unemployment to
employment each quarter. This measure of job creation is used in this Note. The data are from the Household
Labour Force Survey conducted by Statistics New Zealand and are available at a quarterly frequency.

———
1 A dummy variable to control for the impact of the GFC was included initially, however it was insignificant in all specifications tested.

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For job-seekers, the majority of papers surveyed by Petrongolo and Pissarides (2001) simply use the number of
unemployed people. By definition, unemployed people in New Zealand are part of the labour force and are
looking for work, therefore they should be a reasonable proxy for job seekers.2 Unemployment data is also
sourced from the Household Labour Force Survey.

The number of vacancies in New Zealand is proxied using the All Vacancies Index produced by the Ministry of
Business, Innovation, and Employment. These data are timely, but are only readily available from 2010. The
discontinued ANZ job advertisement data are used to backdate the All Vacancies Index to 2000 Q1 (see
Appendix A).

Data are seasonally adjusted using the X-13 ARIMA SEATS programme developed by the US Census Bureau.

B.3: Unit root tests
As noted above, OLS estimates of the ARDL model will be consistent and provide valid standard errors for
inference if variables are integrated of order zero or one. It is therefore important to establish the order of
integration of each variable included in the matching function estimation.

Augmented Dickey-Fuller tests indicate the log-levels of unemployment to employment transitions ( ), number
of unemployed ( ), and vacancies ( ) are all nonstationary (table B.3.1). However, the null hypothesis of a unit
root is rejected for first-differences of each variable. All the variables are integrated of order one over the
sample period of 2000Q1 to 2019 Q4, therefore we can proceed with estimating the ARDL model.

B.4: ARDL model baseline results
The ARDL model of the matching function is estimated from 2000Q1 to 2019Q4 using data on unemployment to
employment transitions, the stock of unemployed people, and the All Vacancies Index spliced together with the
ANZ job advertisements data. The analysis is carried out in Stata, using the ardl package developed by
Kripfganz and Schneider (2018).

 Table B.3.1: Augmented Dickey-Fuller test results

 Variable Test statistic Constant Drift Trend Lags
 -1.652 X 9
 ∆ -3.448*** 2
 -2.080 X X 8
 ∆ -3.560*** 7
 -3.056 X X 3
 ∆ -4.048*** 7
 Note: Schwert criterion is used to select maximum lags for Dickey-Fuller regressions (11 for each variable), then
 testing down until lags are significant at the 10 percent level.
 *** p

For the current analysis, the Bayesian information criterion (BIC) was used to select the lag length. Then,
additional lags of the dependent and independent variables were added to ensure that residuals were not
serially correlated.

The model specification is an ARDL(1,2,1), with one lag of hires (the dependent variable), two lags of the
vacancy index, and one lag of the number of unemployed. A battery of residual diagnostic tests finds evidence
favouring the conclusion that the regression residuals are not serially correlated. The results of the ARDL
model, including residual diagnostics, are outlined in Table B.4.1.

The estimated coefficient on the linear time trend is negative and significant (although only at the ten percent
significance level). This indicates there has been a decline in the efficiency of the New Zealand labour market
since 2000, consistent with findings in other countries (Petrongolo and Pissarides, 2001). The decline in
matching efficiency indicates the labour market has become worse at matching unemployed job-seekers with
vacant jobs, reducing the productivity of the matching function in New Zealand at creating jobs. This decline is
consistent with the outwards shift observed in the Beveridge curve outlined in Section 2.

The estimated long run coefficients on vacancies and unemployment are reported in table B.4.2. These are
obtained from estimating the error correction representation of the ARDL model. The long run coefficients on
vacancies and unemployment are positive and statistically significant at the one percent significance level. The
bounds test confirms there is a long run equilibrium relationship between hires, vacancies, and unemployment
that is statistically significant at the one percent level.3

Since the regression is run in logs, we can interpreted the estimated long-run coefficients on vacancies and
unemployment as elasticities. Specifically:

 A one percent increase in the online vacancy index is associated with a 0.23 percent increase in hires.

 A one percent increase in the number of unemployed job seekers is associated with a 0.58 percent increase
 in hires.

Finally, I fail to reject the null hypothesis that the sum of the coefficients on vacancies and unemployment is
equal to one, indicating the matching function in New Zealand has constant returns to scale.4 This is in contrast
to Razzak (2008) who found decreasing returns to scale for the New Zealand matching function.5

———
3 The null hypothesis of no levels relationship is rejected at the 1 percent significance level. The F-statistic is 28.509, which is above the one percent
 upper critical value of 7.962.

4 Similar results are obtained using the Akaike information criterion (AIC) to select the lag length of the ARDL model. However, this model was less
 parsimonious, with the AIC favouring an ARDL(4,3,2), compared to the ARDL(1,2,1) in the baseline specification.

5 Razzak (2008) uses a similar approach to this paper. However, the source of vacancy data is different (only using ANZ job-ads), and the sample
 period is also very different. It could be the case that over this time the returns to scale of the matching function may have changed. In addition, the
 ANZ data is not cleaned to remove duplicates, which could artificially reduce the number of hires per vacancy posted.

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Table B.4.1: ARDL estimation results

 VARIABLES 

 −1 0.0686
 (0.102)
 0.144
 (0.166)
 −1 -0.122
 (0.201)
 −2 0.190**
 (0.0924)
 -0.398**
 (0.167)
 −1 0.942***
 (0.158)
 Trend -0.00231*
 (0.00124)
 Constant -0.517
 (0.612)
 78
 Observations
 R-squared 0.708

 Residual diagnostic tests

 White test (p-val) 0.4415
 Cameron and Trivedi (p-val) 0.2498
 Breusch-Godfrey (p-val) 0.4644
 Portmanteau (p-val) 0.2162
Note: Standard errors in parentheses. Dependent variable is the natural logarithm of hires
(unemployment to employment transitions) each quarter. Independent variables are the natural
logarithms of the number of unemployed people and the spliced online vacancy index (outlined in detail
in Appendix A), a linear trend, and a constant. *** p

One concern is that the regression relies on a vacancy index constructed from two different sources, where one
is a vacancy index and one is a vacancy count. In Appendix C the matching function is estimated using only the
MBIE data, and then separately using just the ANZ data. The results for the ANZ data are almost identical to the
main results reported in table B.4.2. For the MBIE data, the estimated long-run coefficients were larger than the
baseline results in table B.4.2 for both vacancies and unemployment, although I still failed to reject the null
hypothesis of constant returns to scale. The smaller sample size using the MBIE data made the estimates very
imprecise, with vacancies found to be statistically insignificant despite an increase in the estimated coefficient of
around 25 percent. This highlights the benefit of using the spliced online vacancy index to obtain more precise
estimates.

Appendix C: Robustness tests
C.1: Estimation using MBIE All Vacancies Index
In order to estimate the matching function in Appendix B, I had to combine two measures of vacancies in New
Zealand to get the longest possible sample. This section reports the results of running the baseline ARDL(1,2,1)
model using only the MBIE All Vacancies Index, from 2010 Q4 – 2019 Q4. The estimation results are detailed in
Table C.1.1, noting the small sample of just 36 observations.

 Table C.1.1: ARDL model estimated using All Vacancies Index

 (1)
 VARIABLES 

 −1 -0.0474
 (0.146)
 0.589
 (0.626)
 −1 0.228
 (0.931)
 −2 -0.494
 (0.617)
 -0.158
 (0.275)
 −1 1.183***
 (0.247)
 Trend 0.00261
 (0.00828)
 Constant -3.366*
 (1.750)

 Observations 36
 R-squared 0.544

 Note: Standard errors in parentheses. Dependent variable is the natural logarithm of hires
 each quarter, independent variables are the natural logarithms of the MBIE All Vacancies
 Index and the number of unemployed.
 *** p

The estimated long-run coefficients on vacancies are reported in table C.1.2. Column 1 shows the estimated
coefficients from the baseline ARDL model and column 2 shows the estimated coefficients when restricting the
regression to only using the MBIE All Vacancies Index. For the smaller sample, the coefficients on vacancies
and unemployment are larger although the sample size means the estimates are imprecise and the coefficient
on vacancies is statistically insignificant. I fail to reject the null hypothesis of constant returns to scale, which is
unsurprising given the large standard errors.

The results in tables C.1.1 and C.1.2 indicate that running the matching function analysis using only the All
Vacancies Index leads to qualitatively similar results as using the spliced online vacancies index outlined in
Appendix B, although the small sample means the coefficients are imprecisely estimated. The benefits of using
the longer time-series are that firstly, a larger sample leads to more precise estimates of the regression
coefficients and secondly, it allows us to make conclusions about the New Zealand labour market over a much
longer period of time.

 Figure C.1.2: Estimated long-run coefficients on vacancies and unemployment

 (1) (2)
 VARIABLES 

 0.227*** 0.308
 (0.0847) (0.339)
 0.584*** 0.978***
 (0.0757) (0.275)

 Constant returns to scale p-value
 0.2015 0.4492
 ( : + = 1)

 Observations 78 36

 Note: Newey-West standard errors in parentheses. First column shows baseline results from ARDL model with
 spliced online vacancies, second column shows results obtained using MBIE All Vacancies Index as a proxy for the
 number of vacancies.
 *** p

C.2: Estimation using ANZ advertisements data
As a final check, the matching function analysis is carried out using the ANZ job advertisements data, which
gives us a count of the number of vacancies rather than an index. The ANZ data is not cleaned to remove
duplicates and may therefore overstate how many vacancies are posted in a given quarter. The regression is
carried out over the full range of the ANZ job advertisements data – 2000Q1 to 2018Q4. Table C.2.1 shows the
results for the estimated ARDL model.

 Table C.2.1: ARDL model estimated using All Vacancies Index

 (1)
 VARIABLES 

 −1 0.0813
 (0.103)
 0.105
 (0.150)
 −1 -0.0804
 (0.175)
 −2 0.153**
 (0.0744)
 -0.415**
 (0.172)
 −1 0.941***
 (0.161)
 Trend -0.00203
 (0.00127)
 Constant -1.482
 (1.055)

 Observations 74
 R-squared 0.714

 Note: Standard errors in parentheses. Dependent variable is the natural logarithm of hires each
 quarter, independent variables are the natural logarithms of the ANZ job advertisements count and
 the number of unemployed.
 *** p

Table C.2.2 shows the estimated long run coefficients on vacancies and unemployment when using the ANZ job
advertisements data (column 2), compared with the baseline specification (column 1). The estimated long run
coefficient on the ANZ job advertisements variable is very similar to the estimated coefficient for the spliced
online vacancies index. Once again, the null hypothesis of constant returns to scale is not rejected at all
conventional significance levels.

 Figure C.2.2: Estimated long-run coefficients on vacancies and unemployment

 (1) (2)
 VARIABLES 

 0.227*** 0.194**
 (0.0847) (0.0785)
 0.584*** 0.573***
 (0.0757) (0.0796)

 Constant returns to scale p-value
 0.2015 0.1067
 ( : + = 1)

 Observations 78 74

 Note: Standard errors in parentheses. First column shows baseline results from ARDL model with
 spliced online vacancies, second column shows results obtained using ANZ job advertisements
 measure of the number of vacancies.
 *** p

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