S3H Working Paper Series

Number 06: 2019

Non-linear Model of Aggregate Credit Risk for

Banking Sector of Pakistan: A Threshold

Vector Autoregressive Approach

Muhammad Anwaar Alam Khokhar

Ather Maqsood Ahmed

November 2019

School of Social Sciences and Humanities (S3H) National University of Sciences and Technology (NUST)

Sector H-12, Islamabad, Pakistan

S3H Working Paper Series

Faculty Editorial Committee

Dr. Zafar Mahmood (Head)

Dr. Samina Naveed

Dr. Gulnaz Zahid

Dr. Ume Laila

Dr. Ashfaq Shah

Dr. Umar Nadeem

S3H Working Paper Series

Number 06: 2019

Non-linear Model of Aggregate Credit Risk for

Banking Sector of Pakistan: A Threshold

Vector Autoregressive Approach

Muhammad Anwaar Alam Khokhar

Assistant Director, Agricultural Credit & Microfinance Department, State Bank of Pakistan

E-mail: muhammad.anwaar@sbp.org.pk

Ather Maqsood Ahmed

Professor, School of Social Sciences and Humanities, NUST E-mail: ather.ahmed@s3h.nust.edu.pk

November 2019

School of Social Sciences and Humanities (S3H) National University of Sciences and Technology (NUST)

Sector H-12, Islamabad, Pakistan

iii

Table of Contents

Abstract ........................................................................................................................................................ v

1. Introduction ........................................................................................................................................ 1

2. Literature Review ................................................................................................................................ 4

3. Data ...................................................................................................................................................... 9

4. Estimation Methodology ................................................................................................................. 12

4.1 Baseline VAR Model ............................................................................................................... 12

4.2 Non-Linear Threshold VAR Model ...................................................................................... 13

4.3 Determining the Threshold Value ......................................................................................... 14

4.4 Forecasting Methodology ....................................................................................................... 19

5. Results ................................................................................................................................................ 20

6. Conclusion ......................................................................................................................................... 26

7. References .......................................................................................................................................... 27

List of Tables

Table 1 – Descriptive Statistics ................................................................................................................. 9

Table 2 – Unit Root (ADF) Tests .......................................................................................................... 11

Table 3 – Lag Selection Based on Information Criteria ...................................................................... 12

Table 4 – Grid Search for Threshold Specification ............................................................................. 17

Table 5 – Out of Sample Forecast Methodology ................................................................................. 20

Table 6 – Threshold Test for Non-Linearity ...................................................................................... 210

Table 7 – MAPE: In Sample Forecast ................................................................................................. 221

Table 8 – MAPE: Out of Sample Forecast…………………………………………………….22

List of Figures

Figure 1 – Run Sequence Plots of the Variables .................................................................................. 10

Figure 2 – Scatter Plots of the Variables ............................................................................................... 14

Figure 3 – Scatter Plot of Gross NPL Ratio against Threshold Variable ......................................... 18

Figure 4 – Regression RMSE vs. Threshold Variable ......................................................................... 18

Figure 5 – Run Sequence Plots of Residuals of the Models ............................................................... 21

Figure 6 – Out of Sample Forecast around the Historical Rise and Peak ........................................ 23

Figure 7 – Threshold VAR Model: Generalized Impulse Responses of Gross NPL Ratio to +/-

1&2 SD Shocks to All Variables ........................................................................................ 24

Figure 8 – Linear VAR Model: Impulse Response of Gross NPL Ratio to +1 SD Shock to All

Variables ................................................................................................................................. 25

iv

Acronyms

AMG Augmented Mean Group

APE Average Percentage Error

ARDL Autoregressive Distributed Lag

BCBS Basel Committee on Banking Supervision

CAR Capital Adequacy Ratio

CPI Consumer Price Index

CPV Credit Portfolio View

FSR Financial Stability Review

GDP Gross Domestic Product

GFC Global Financial Crisis

GIRF Generalized Impulse Response Function

GNPLR Gross Non-Performing Loans Ratio

INF Inflation

IRF Impulse Response Function

LSM Large Scale Manufacturing

LSMG Large Scale Manufacturing Growth

MA Moving Average

MAPE Mean Absolute Percentage Error

NPL Non-Performing Loans

RMSE Root Mean Squared Error

RR Reverse Repo Rate

RWA Risk Weighted Assets

SBP State Bank of Pakistan

TVAR Threshold Vector Autoregression

VAR Vector Autoregression

VaR Value at Risk

v

Abstract

The Global Financial Crisis highlighted the failure on part of risk managers and regulators to adequately

determine and account for the buildup of risks in the financial systems. It also highlighted the inherent inadequacy of

conventional models to capture and predict the tail risks in the financial sector and provide reliable forecasts under stressed

scenarios. Much recently, focus has shifted towards building innovative models to account for these risks. This study

develops a Threshold Vector Autoregressive model for the banking sector of Pakistan and compares its accuracy to

conventional linear counterpart in terms of forecasting Gross Non Performing Loans ratio, a key financial stress

indicator. The results suggest the presence of a significant threshold in the data generating process and the estimated

threshold model as faring better at predicting the Gross Non-Performing Loans ratio with much lower forecasting errors

for up to four period ahead forecasts, particularly at longer horizons.

JEL Codes: C32; C53; E58; G21; G32

Keywords: credit risk, non-performing loans, stress test, non-linear model, threshold, VAR

forecasting, generalized impulse response

1

1. Introduction

The study falls under the domain of financial stability analysis, which has gained much

importance after the Global Financial Crisis (GFC) of 2007-08, albeit carried out even before that.

The collapse of the subprime housing bubble in the United States led to the failure of many Global

Systemically Important Financial Institutions and through cross border contagion quickly spread to

the whole world. The global economy took almost a decade to recover from the aftermath of the

financial crisis. A failure on the part of risk managers and regulators to identify the possible burst of

the housing market bubble, and more importantly, to identify the presence of such high levels of

lending concentrations and interconnectedness of the financial institutions ultimately led to the crisis

getting out of hand and huge bailout plans becoming necessary for recovery. This further highlighted

the importance of stress testing the financial systems to identify the systemically important financial

institutions, and to keep a vigilant watch over the shortcomings and weak points in the system that

could trigger a crisis following a significant exogenous shock (Dent, Westwood, & Segoviano, 2016).

The GFC also took a toll on the economy of Pakistan through channels such as decline in

capital inflows due to general uncertainty, rise in international fuel and food prices, and most

importantly, the sudden severe decline in the international trade which affected many emerging

economies heavily. The local policy responses to the adverse national and international scenario

resulted in high inflation and rising twin deficits coupled with drastic decline in economic growth. In

turn, the local financial system also suffered a stressful period with rising defaults and liquidity issues

(Shabbir, 2010).

The Basel Committee on Banking Supervision (BCBS) provides high-level principles for the

regulation of banking systems. The principles are non-binding; however, they are widely adopted by

central banks and regulatory authorities across the world. The committee published Basel I accord in

1988, which focused primarily on the credit risk and proper risk weighting of assets, and therefore,

the capital adequacy of banks (Basel Committee on Banking Supervision, 1988). The Basel II is the

second accord published by BCBS in 2004, which was implemented by regulatory authorities in years

before 2008. This accord provided a three-pillar approach encompassing principles for minimum

capital requirements, supervisory review, and market discipline. Under the capital requirements, apart

from credit risk, market and operational risks were also introduced, while the credit risk framework

was also strengthened (Basel Committee on Banking Supervision, 2004). The Basel III accord was

agreed upon in 2010 but the implementation started from 2013 in a phased manner up to 2019. Post

2

Global Financial Crisis, the latest accord is intended to strengthen bank capital requirements through

increasing liquidity and decreasing leverage. The accord also shifted the focus from micro prudential

to macro prudential supervisory framework and introduced capital conservation and counter cyclical

buffers for ensuring capital adequacy in stressful times, while easing banks’ stress. One of the key

capital requirements is the Capital Adequacy Ratio (CAR), which is the ratio of bank’s total eligible

capital to the total risk weighted assets (RWA) (Basel Committee on Banking Supervision, 2010). The

State Bank of Pakistan implemented Basel III starting from 2013 in a phased manner up to 2019. The

minimum Total Capital to RWA ratio is kept at 10% throughout this phase. However, banks are also

required to hold a capital conservation buffer from 2015 onwards in a phased manner. By the end of

2019, in the event of full implementation of Basel III, the minimum CAR requirement will be 12.5%

including the capital conservation buffer (State Bank of Pakistan, 2013).

In the context of the aforementioned Basel and regulatory requirements, from a regulatory

point of view, it is important to be able to predict whether a bank, or the system as a whole, has

sufficient capital to withstand exogenous macroeconomic shocks, or whether a potential economic

downturn will diminish the bank capital below the minimum requirements.

To this end, the regulatory bodies regularly conduct macro financial stress tests. Stress testing

is a technique used by regulatory authorities and individual institutions to run a stressed scenario

analysis to predict whether an institution, or the financial system as a whole, will be able to cope with

extreme but plausible shocks. In other words, stress tests determine the level of shock/stress an

institution or the system can bear without triggering a failure (of an individual institution) or a crisis

(of the overall financial system). This is particularly useful for regulators and central banks since such

tests provide a benchmark to assess the health of the system and the individual institutions for timely

policy interventions (Bunn, Cunningham, & Drehman, 2005).

Macro financial stress testing involves three stages; (i) forecasting of macroeconomic variables

to develop baseline (well performing economy) and stressed scenarios; (ii) feeding these forecasts into

a model to forecast a stress indicator (gross non-performing loans ratio, write off ratio etc.); (iii)

mapping the forecasted stress indicator onto the banks’ capital to determine the future Capital

Adequacy Ratio (Blaschke, Jones, Majnoni, & Peria, 2001). In the third stage, if a bank’s CAR is

predicted to go below the minimum requirements then the bank is said to be financially stressed and

adequate measures are taken pre hand to improve the financially stressed bank’s capital.

Among the linkages between macroeconomic factors and NPLs, the widely used variables that

relate macroeconomic factors to non-performing loans are inflation, interest rates, and economic

3

growth, among others. Many studies have found strong linkages between these variables and NPLs.

For example, Endut et al. (2013) found significant positive long run impact of inflation on NPLs and

a negative relationship between Gross Domestic Product and NPLs. Moreover, interest rates were

found to positively impact NPLs. Their findings parallel those of earlier studies such as Nkusu (2011),

Espinoza & Prasad (2010) etc. The theoretical business cycle models with an explicit role for financial

intermediation offer a good background for modeling NPL as they highlight the countercyclicality of

credit risk and business failures (Williamson, 1987). As such, inflation is expected to have a positive

impact on NPLs as inflation reduces real income and also lowers collateral value thereby rendering

repayment more difficult. The same relation is expected from interest rate as a rise in interest rate

directly relates to higher repayment in case of variable interest loans. Moreover, economic growth is

argued to affect NPLs negatively as businesses find it conducive to pay back the loans from higher

economic activity and greater returns on investments (Nkusu, 2011).

Among the varying approaches to model the stress indicator, is the widespread use of Vector

Autoregressive models, primarily for their simplicity and ability to use different scenarios to construct

forecasts. However, the use of simple linear models has been criticized for their inability to capture

the tail risks in the system (Drehmann, Patton, & Sorensen, 2007). Therefore, focus has shifted

towards developing innovative models including non-linear specifications to capture the tail risks

adequately and provide forecasts that are more reliable.

A review of the literature for Pakistan suggests that there exists a gap in literature regarding

empirical studies involving complex models for the determinants of non-performing loans, which may

be utilized to stress test the banking system. This study, in an attempt to cover the gap, utilizes a non-

linear methodology, i.e., the Threshold Vector Autoregressive model on the data for the banking

sector of Pakistan in an attempt to adequately capture the data generating process and develop more

accurate forecasts and non-linear impulse responses in comparison to the linear models, as required

under step (ii) of stress testing methodology outlined previously. A top down approach focusing on

the aggregate system level data is employed. The financial stress is modeled with Gross Non

Performing Loans ratio of the overall banking industry as an indicator. In essence, this study provides

an overall macroeconomic analysis of the financial system as a whole and its responses to shocks both

to the financial sector and to the macro economy. The non-linear threshold specification of Vector

Autoregressive model (TVAR) is particularly useful in generating non-linear impulse responses that

are sensitive to both size and sign of the shock, and dependent on the initial conditions of the system

(Koop, Pesaran, & Potter, 1996).

4

The central objective of the study is to determine the presence of threshold effects in the

subject dataset and estimate a significant threshold, if any, and provide a forecasting model in an

attempt to improve forecast accuracy compared to a linear VAR alternative.

The analysis carried out in the study is based on four variables: inflation rate, interest rate

(State Bank of Pakistan reverse repo rate), Large Scale Manufacturing (LSM) growth rate, and the

gross non-performing loan (NPL) ratio, without the deduction of loan loss provisions. A quarterly

dataset is employed from the period 2006:Q1 to 2018: Q1.

The study is structured as follows; in section 2, the existing theoretical literature and major

contributions in this field are reviewed, both specifically for the case of Pakistan as well as for other

countries. In section 3 the dataset is discussed and graphical representations of the data are analyzed

for possible presence of non-linearity. In section 4, the non-linear Threshold VAR model and the

estimation methodology is discussed in detail and the chosen specifications are highlighted and

explained in light of the theoretical and empirical contributions in the past. Also, the different

techniques used for constructing the forecasts and the non-linear impulse response functions are

discussed and explained in light of econometric theory and the dynamics of the dataset at hand. In

section 5, the results are presented and discussed. Finally, section 6 concludes the study, highlights

the shortcomings, provides relevant policy recommendations, and identifies possible areas for further

analysis.

2. Literature Review

An extensive volume of research has been carried out in the domain of financial stability. From

bottom up approaches, focusing on individual banks to reach the aggregate stability of the sector, to

top down approaches focusing on the system vide aggregate data for the analysis. These include

individual contributions of researchers as well as analysis carried out by central banks and other

regulatory bodies.

In the context of Pakistan, the State Bank of Pakistan (SBP) has a dedicated Financial Stability

Department to monitor and analyze the health of the aggregate financial sector and to carry out regular

stress tests to identify potential problems in the financial institutions. The leading publication of the

department; Financial Stability Review (FSR), published annually, contains the top down stress test of

the country’s financial sector. In the previous edition, FSR 2016, a linear VAR model is used where

Gross Non Performing Loan Ratio (GNPLR) is used as the financial stress indicator and is considered

a function of industrial output, exports, developments in stock market, inflationary pressure, and

5

prevailing risk pricing (State Bank of Pakistan, 2016). Stress scenarios of both hypothetical and

historical nature are designed and quantitative forecasts of sector specific key variables, based on

historical domestic and global shocks, are used as inputs for the stress tests. The analysis, however, is

only carried out based on linear models and no non-linear specification is employed. The latest version,

Financial Stability Review of 2017, extends the study in terms of more innovative and complex

scenario analysis; however, the models employed for stress testing are essentially the same (State Bank

of Pakistan, 2017).

Moreover, for the banking sector of Pakistan, Ahmad & Bashir (2013) find the determinants

of NPLs based on annual data from 1990 to 2011. They find that among the nine employed

macroeconomic variables, GDP growth, interest rate, inflation rate, CPI, exports, and industrial

prodcution are the significant determinants of NPLs, where inflation rate was found to have a positive

impact on NPLs while the other variables were found to affect NPLs negatively. In another study, a

quadratic relationship was found between inflation and default rate (Rizvi & Khan, 2015). The analysis

was, however, carried out with net NPLs which does not truly reflect the default rate as the loan loss

provisions are deducted from gross NPLs to reach a net figure. Mahmood (2018) employed a panel

data approach to find the determinants of NPLs using data from 39 Pakistani banks, allowing for bank

specific and macroeconomic factors. It was found that among the macroeconomic factors, real lending

rate has a significant positive impact on NPLs. However, GDP and unemployment rate were found

to be insignificant in the analysis. Jameel (2014) developed a multiple linear regression model for the

determinants of NPLs in Pakistan based on 11 year annual data. Among macroeconomic factors,

GDP was found to have a significant negative impact on the NPLs and weighted average lending rate

was found to be significantly positively related to NPLs.

For the Turkish economy, stress test has been carried out using the Credit Portfolio View

(CPV) Model based on an unrestricted VAR of linear specification (Basarir, 2016). A satellite model

was constructed using quarterly dataset between 1999 and 2012, NPLRE (non-performing loan index)

derived by logit transformation of NPLR series, and a set of macroeconomic variables. Scenarios were

created based on historical shocks to interest rate, and/or exchange rate. It was concluded, based on

these scenarios, that the Turkish banking sector was highly resilient to shocks similar to 2001 crisis.

For the Hong Kong banking sector, Wong et al. (2006) develop a stress test model based on

the methodologies presented in Wilson (1998) and Virolainen (2004). Quarterly dataset between

1994:Q4 and 2006:Q1 is used in this study. The default rate is explained using a set of macroeconomic

variables: real GDP growth of Hong Kong, real GDP growth of Mainland China, real interest rates in

6

Hong Kong, and real property prices in Hong Kong. The model is estimated using Seemingly

Unrelated Regression technique and four historical scenarios are constructed based on adverse shocks,

observed during the Asian Financial Crisis, to the four macroeconomic variables considered in the

model. It was concluded that the considered set of data shows that the retail banks are resilient to

moderate levels of these shocks and continue to profit, however, they could incur losses at very high

but extremely unlikely shock levels.

Hoggarth et al. (2005) use a multidimensional approach to stress testing the banking sector of

UK. They have included both a bottom up approach as well as an aggregate top down approach. A

VAR model is also estimated based on the write-off ratio (write-offs to total loans) and on explanatory

macroeconomic variables; output relative to trend (Output Gap), nominal short-term interest rate,

real exchange rate, and inflation rate. A linear specification is chosen for the analysis. Stressed scenarios

are designed based on shocks to the macroeconomic explanatory variables. It was concluded that a

range of plausible but extreme shocks have no significant impact on the stability of the banking sector

of UK.

For the Brazilian banking sector, Vazquez et al. (2012) develop two models. One involves

simulation of bank level NPLs under distressed scenarios but without allowing for differences in credit

quality across credit types. The other model allows for such differences to be present. The approach

taken is threefold. Firstly, they use a macroeconomic VAR model to forecast under distressed

scenarios. Secondly, they use these forecasts as inputs to their microeconomic, bank-level, panel

models (one with and one without allowing for differences across credit types) to gauge the path of

NPLs based on the adverse scenarios. Thirdly, they use a credit Value-at-Risk (VaR) approach to

estimate banks’ capital needs to cover tail end losses arising from these scenarios. Their results

highlight a need for better-fit models, and suggest that a portfolio aggregation bias may exist in many

existing studies as, in the top down approach; highly aggregated data is used whereas credit quality

may actually differ across different credit types.

Grigoli et al. (2016) employed a similar multi-stage methodology for the Ecuadorian banking

sector. In the first stage, a Structural VAR model is estimated for forecasting the key variables. Then

in the second stage, using bank level panel dataset of NPLs, these forecasts are fed into a Panel data

Autoregressive Distributed Lag (ARDL) model where logistic transformation of NPL series is kept

the dependent variable and to cater for the dynamics of the dependent variable an adjustment

parameter is incorporated in the said model. This is estimated using Augmented Mean Group (AMG)

technique and bank specific forecasts of NPLs are estimated and averaged. What significantly

7

differentiates their study from the rest is the selection of the data series for the modeling. They use oil

price as a proxy for liquidity in the financial sector through exports and fiscal revenues. In addition,

they argue that since Ecuador follows a full dollarization regime, the interest rates are capped and

therefore market interest rates/yield curve, and/or exchange rate cannot be effectively used to account

for the monetary policy shocks. Hence, a different set of variables; real GDP, real supply of credit,

and real deposits, is used along with oil price as an exogenous variable to introduce shocks in the

system. Their results suggest that significant heterogeneity exists across banks; however, a one

percentage point fall in real GDP growth can double the weighted average NPL ratio for the financial

system as a whole over a two year period.

The threshold model in economics was first introduced in Tong (1978). The initial works are

based on univariate autoregressive models classified as Threshold Autoregressive (TAR) models.

These models have been extensively used in literature, both for further developments and for practical

applications. The threshold value in the simple TAR model may be based on any variable that may or

may not directly go into the estimation. However, for the models where the threshold is directly

determined through the model’s variable itself are classified under the class of Self-Exciting Threshold

Autoregressive (SETAR) models. Further works in developing the model and tests for selection of

threshold come from works such as Tong and Lim (1980), Tsay (1989), and Tong (1990) etc.

Moreover, the works by B. E. Hansen add to the literature regarding inference of the TAR models

and the tests for presence of threshold. For example, Hansen (1996) provides inference on the tests

for significance of the threshold when the nuisance parameter is not identified under the null

hypothesis. Hansen (1999) provides a methodology to test the presence of non-linearity in the data

and to accurately fit a model accordingly. These concepts have been further developed into more

complex models. For example, the Smooth Transition Autoregressive (STAR) models enable to

specify for a smooth transition between regimes when the threshold variable crosses the threshold

value instead of the abrupt switch that the simple TAR or SETAR models provide. The significant

contributions in this class of models comes from Chan and Tong (1986), Teräsvirta and Anderson

(1992), Teräsvirta (1994), and Eitrheim and Teräsvirta (1996), among others.

The threshold modeling has also been incorporated into the much established and widely used

Vector Autoregressive models introduced by Sims (1980). For example, Tsay (1998) provides

modeling techniques for Threshold VAR models. Further developments include Tena and Tremayne

(2009) where the assumption of a single threshold variable for the whole system is relaxed and each

equation in the system is allowed to have a threshold independent of the other equations. More

8

applications include Sims and Zha (2006), Sims et al. (2008), and Hubrich and Tetlow (2014) among

others. Anderson and Vahid (1998) incorporate smooth transitions into the vector class of models

developing Logistic Vector Smooth Transition Regression model. More recent applications include

Terasvirta and Yang (2014).

For non-stationary time series analysis, thresholds have been incorporated into the Vector

Error Correction models (VECM). For example, Balke and Fomby (1997) extended the Threshold

VAR model for cointegrating relationships. Moreover, VECM models have been extended to include

smooth transitions as well. See for example, Rothman et al. (2001), Camacho (2004), and Goodwin et

al. (2012).

For the case of the threshold VAR model, as employed in this study, the tests for linearity are

based on calculation of Wald statistics testing the coefficients to significantly differ between a linear

and non-linear specification. However, because of the threshold variable being a nuisance parameter

in the estimation, the resulting statistic is not operational. To cater for this issue, a number of statistics

are calculated. More specifically, by calculating the statistic over a range of the threshold variable, after

removing sufficient buffer from each extreme of the variable’s range, a supremum and an average

Wald statistic is calculated. Hansen (1996) provides simulation and bootstrap methodology to

construct the empirical distribution function for inference of these statistics. Studies by Balke (2000)

and Atanasova (2003) incorporate the methodology into the vector class of models.

Koop et al. (1996) provide the framework for constructing impulse responses through a non-

linear model. They classify such impulse responses as Generalized impulse response functions

(GIRFs). The impulse responses through a non-linear model are not proportional to the size of the

shock. Further, they also depend on the history, or the initial conditions from where the impulse

responses are constructed. For example, a Threshold VAR model with two regimes would be able to

provide GIRFs where; (i) different sized shocks have different responses; (ii) positive and negative

shocks may not be necessarily symmetrical; and (iii) the impulse responses will be different in the two

regimes. These GIRFs are highly useful in drawing inference from the model since interpreting the

coefficient estimates is not possible in itself.

The review of the subject for the case of Pakistan reveals that a number of studies have been

conducted in the past, varying among methodologies and choice of variables, however, threshold

models have not been applied to the subject before. Moreover, the use of threshold models for similar

studies in case of other countries further warrants deeper insights into the subject for the case of

Pakistan.

9

3. Data

The data set used for this study is quarterly and spans from 2006Q1 to 2018Q1. The variables

employed are gross non-performing loans to total loans ratio (GNPLR), year on year inflation (INF),

SBP Reverse repo rate (RR), and year on year growth rate of Large Scale Manufacturing (LSMG). The

data has been sourced from quarterly publications of State Bank of Pakistan. Table 1 provides the

descriptive statistics for the variables. The data set contains 49 observations of all variables. The lowest

observed value of GNPLR is 0.071 while it peaks at 0.167 with a mean of 0.116. The INF series has

a minimum of 0.018 while it peaks at 0.261 with a mean of 0.090. The RR series minimum is 0.063

while maximum is 0.150. The LSMG series has a mean of 0.034 while the minimum and maximum

are -0.082 and 0.143 respectively. The GNPLR and RR series have lower standard deviation of 0.029

and 0.027 respectively, however, the standard deviation of INF and LSMG are higher i.e. 0.054 and

0.052 respectively.

Figure 1 provides the run sequence plots of the variables. It can be seen that Inflation remained

somewhat stable up to 2007Q4 when it started rising sharply and reached its maximum in 2008Q3

(panel a). As a response to the sharp rise in inflation, SBP rate also started to rise at the same time

(panel b). LSM growth also declined sharply following the rise in inflation, and reached its minimum

in 2008Q4 (panel c). All these factors contributed in hampering the repayment capacity of bank

borrowers and therefore resulted in the rise of non-performing loans of banks. As may be seen in

figure 1(d), GNPLR reached its minimum in 2007Q2 and then started rising and reached the

maximum in 2011Q3.

Table 1. Descriptive Statistics

GNPLR INF RR LSMG

Mean 0.116 0.090 0.101 0.034

Median 0.122 0.081 0.100 0.041

Maximum 0.167 0.261 0.150 0.143

Minimum 0.071 0.018 0.063 -0.082

Std. Dev. 0.029 0.054 0.027 0.052

Skewness -0.052 1.339 0.077 -0.241

Kurtosis 1.743 4.882 1.912 2.527

Jarque-Bera 3.247 21.875 2.463 0.930

Probability 0.197 0.000 0.292 0.628

Sum 5.681 4.388 4.933 1.686

Sum Sq. Dev. 0.041 0.141 0.036 0.128

Observations 49 49 49 49

10

Figure 1. Run Sequence Plots of the Variables (a) – Inflation Rate

.00

.05

.10

.15

.20

.25

.30

2006:Q1

2007:Q1

2008:Q1

2009:Q1

2010:Q1

2011:Q1

2012:Q1

2013:Q1

2014:Q1

2015:Q1

2016:Q1

2017:Q1

2018:Q1

Inflation

(b) – SBP Reverse Repo Rate

.06

.08

.10

.12

.14

.16

2006:Q1

2007:Q1

2008:Q1

2009:Q1

2010:Q1

2011:Q1

2012:Q1

2013:Q1

2014:Q1

2015:Q1

2016:Q1

2017:Q1

2018:Q1

SBP Reverse Repo Rate

(c) – LSM Growth Rate

-.10

-.05

.00

.05

.10

.15

2006:Q1

2007:Q1

2008:Q1

2009:Q1

2010:Q1

2011:Q1

2012:Q1

2013:Q1

2014:Q1

2015:Q1

2016:Q1

2017:Q1

2018:Q1

LSM Growth

11

(d) – Gross Non Performing Loan Ratio

.06

.08

.10

.12

.14

.16

.18

2006:Q1

2007:Q1

2008:Q1

2009:Q1

2010:Q1

2011:Q1

2012:Q1

2013:Q1

2014:Q1

2015:Q1

2016:Q1

2017:Q1

2018:Q1

Gross NPL Ratio

The variables are then tested for presence of unit roots. The run sequence plots in figure 1

suggest that all the series contain breaks and/or nonlinear trend. In such a case, ADF test would not

be appropriate, instead the unit root test with break points, as proposed by (Perron, 1989) is more

suitable. The specification for the tests for all variables includes trend and intercept with a break in

trend. The null hypothesis of the test is that the variable contains a unit root against the alternative

that the series is trend stationary with a break point. Table 2 provides the results of the tests. It is

found that at 5% significance level, all the variables are trend stationary at level with their respective

breakpoints.

Table 2. Breakpoint Unit Root Tests

GNPLR INF RR LSMG

Level

-4.544720 -5.907722 -5.183325 -4.931905

(0.0478)* (<0.01)* (<0.01)* (0.0159)*

Breakpoint 2012Q1 2008Q4 2010Q3 2009Q2

Note: Unit root tests; p-values in parenthesis, * significant at 5%.

Conventional unit root tests are also criticized for their low power and bias in favor of rejecting

the null when there is non-linearity in the data particularly in presence of threshold effects. For

example, Pippenger and Goering (1993), Balke and Fomby (1997), and Taylor (2001). To cater for

these inadequacies, many non-linear unit root tests have also been developed. For example, Gonzalez

and Gonzalo (1997), Enders and Granger (1998), Caner and Hansen (2001), Bec et al. (2004) and

12

Kapetanios and Shin (2006). However, testing for threshold unit root involves various complexities

and, therefore, goes beyond the scope of this study.

4. Estimation Methodology

Developing a non-linear model to adequately capture the non-linearity in the data can be

accomplished through various approaches and does not have a specific sequence that may be followed,

as otherwise in the case of linear models. Further, to gauge the performance of the non-linear model,

a linear model needs to be estimated first to establish a baseline against which to compare the non-

linear model. Therefore, as a first step, a baseline Vector Autoregressive model is fitted to the data,

and then the assumptions are relaxed and necessary adjustments are made to find a suitable threshold

and fit a threshold model.

4.1 Baseline VAR Model

Although the variables are found to be non-stationary, for the baseline model, the variables

are selected in levels. Optimal lag length is selected based on information criteria for up to four lags.

Except SIC, all other information criteria suggest a lag length of two (Table 3).

Table 3. Lag Selection Based on Information Criteria

Lag LogL LR FPE AIC SC HQ 0 321.36 NA 4.17e-13 -17.15 -16.98 -17.09 1 446.97 217.28 1.12e-15 -23.08 -22.21* -22.77 2 474.43 41.56* 6.24e-16* -23.70* -22.13 -23.15* 3 485.97 14.96 8.65e-16 -23.46 -21.19 -22.66 4 498.92 14.01 1.21e-15 -23.29 -20.33 -22.25

* indicates lag order selected by the criterion LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion Lag length selection.

Therefore, a linear VAR in levels model of the form below with a lag length of two is then estimated

and the linear impulse response functions are constructed as a baseline.

13

𝑌𝑡 = 𝛼0 + ∑ 𝛼𝑖

2

i=1

𝑌𝑡−𝑖 + 𝜀𝑡

where, the linear vector 𝑌𝑡 = [𝐺𝑁𝑃𝐿𝑅𝑡 , 𝐼𝑁𝐹𝑡, 𝑅𝑅𝑡, 𝐿𝑆𝑀𝐺𝑡]

4.2 Non-Linear Threshold VAR Model

Threshold model is a kind of a regime switching model whereby regime switches are

determined by values of a certain variable crossing a threshold (in case of Threshold Models) as

opposed to being determined by time (in case of Regime Switching/Structural Break models). These

threshold models range from simple threshold autoregressive models (TAR) to complex VAR models

with threshold effects (TVAR & THSVAR). In case of univariate models, the threshold may be

determined through the same variable, or through some other variable that does not directly go into

the regression. If the variable itself determines the threshold then such a model is called Self Exciting

Threshold Autoregressive Model (SETAR). Some even more complex models exist in the literature,

where it is argued that the switch between regimes does not happen abruptly, but instead the model

parameters go through a smooth transition when the threshold is crossed. Such models fall under the

class of Smooth Transition Threshold Autoregressive (STAR) models. Smooth transitions have also

been incorporated into VAR models (STVAR).

For this study, a Threshold Vector Autoregressive (TVAR) model is developed to capture the

non-linearity in the underlying data generating process. The model can be expressed as follows:

𝑌𝑡 = 𝛼0 + ∑ 𝛼𝑖

p

i=1

𝑌𝑡−𝑖 + 𝑈𝑡 If 𝐶𝑡−𝑑 > 𝛾

𝑌𝑡 = 𝛽0 + ∑ 𝛽𝑖

p

i=1

𝑌𝑡−𝑖 + 𝑉𝑡 If 𝐶𝑡−𝑑 ≤ γ

where, Y is a vector of variables, C is the threshold variable which may or may not belong to Y, d is

the delay parameter (explained later), and is the threshold value.

Alternatively, the model may be expressed as:

𝑌𝑡 = (𝜃1 ∑ 𝑌𝑡−𝑖

p

i=1

)𝐼(𝐶𝑡−𝑑 ≤ 𝛾) + (𝜃2 ∑ 𝑌𝑡−𝑖

p

i=1

)𝐼(𝐶𝑡−𝑑 > 𝛾) + 𝜀𝑡

where, the indicator functions 𝐼(𝐶𝑡−𝑑 > 𝛾) equals 1 when 𝐶𝑡−𝑑 > 𝛾 and 0 otherwise, and

𝐼(𝐶𝑡−𝑑 ≤ 𝛾) equals 1 when 𝐶𝑡−𝑑 ≤ 𝛾 and 0 otherwise.

14

4.3 Determining the Threshold Value

The determination of the threshold is the most critical element in the estimation, as there does

not exist a standard methodology that may be employed to determine the threshold value. As an initial

step, scatter plots of the dependent and independent variables can be analyzed to check for any signs

of non-linearity. Figure 2 provides scatter plots of Gross NPL Ratio against Inflation, SBP Reverse

Repo Rate, and LSM Growth.

Figure 2. Scatter Plots of the Variables (a) – Gross NPL Ratio vs. Inflation Rate

.06

.08

.10

.12

.14

.16

.18

.00 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28

Inflation

Gro

ss N

PL R

atio

Linear Fit

Power Fit

Polynomial Fit

(b) – Gross NPL Ratio vs. SBP Reverse Repo Rate

.06

.08

.10

.12

.14

.16

.18

.06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16

SBP Reverse Repo Rate

Gros

s NPL

Rat

io

Linear Fit

Power Fit

15

(c) – Gross NPL Ratio vs. LSM Growth Rate

.06

.08

.10

.12

.14

.16

.18

-.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12 .14 .16

LSM Growth

Gro

ss N

PL R

atio

Linear Fit

Power Fit

(d) – Inflation vs. LSM Growth Rate

.00

.05

.10

.15

.20

.25

.30

-.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12 .14 .16

LSM Growth

Infla

tion

Linear Fit

Polynomial Fit

(e) – SBP Reverse Repo Rate vs. LSM Growth Rate

.06

.08

.10

.12

.14

.16

-.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12 .14 .16

LSM Growth

SBP

Reve

rse

Repo

Rat

e

Linear Fit

Polynomial Fit

16

In each panel, Linear and Power/Polynomial Fit lines are provided. The Power Fit line is the

regression line with quadratic transformation of the dependent variable whereas the Linear Fit line is

the regression line with no transformation of the dependent or independent variables. As can be seen,

the scatter plot of Gross NPL Ratio vs. Inflation (panel a) shows a slightly positive relation with

negligible difference between linear and power fit lines, however, a polynomial fit shows non-linearity

but the possible break corresponds to outliers in the data. The scatter plot of Gross NPL Ratio vs

Interest Rate (panel b) also depicts an increasing relationship between the variables with negligible

difference between the linear and power fit lines. However, in the Gross NPL Ratio vs LSM Growth

plot (panel c), a clear non-linear relationship is evident. The linear fit line shows a continuous declining

relationship between the variables whereas, the power fit suggests that as LSM Growth initially

increases Gross NPL Ratio also increases but after a certain threshold the relationship between the

variables becomes negative. In panels (d) and (e), a nonlinear relationship is seen between inflation

and LSM growth, and SBP Reverse Repo rate and LSM growth rate, respectively. The polynomial line

provides more accurate fit to the data compared to the linear fit and a change in slope is witnessed

around 0.03 value of LSM growth in both plots.

To further determine the presence of a threshold in the regression, a different approach is

adopted which resembles the methodology presented in Balke (2000). Based on the scatter plots of

Figure 2, the testing is done for presence of threshold through LSMG series. The threshold variable

(𝐶𝑡−𝑑) selected for this study is the 2 period moving average of the underlying variable (in this case

LSM Growth). The length of moving average of 2, and the delay parameter (d) equal to 1 are selected

based on a grid search methodology. This means that, for example, for the period 2008Q3, the regime

will be determined by the average of LSM Growth for the periods 2008Q1 and 2008Q2. Further, there

may be more than a single threshold in the model such that there may be more than two regimes. The

number of possible thresholds does not have a limit, but general estimation limitations do apply. For

example, for small data sets, it would not be suitable to find multiple thresholds and have insufficient

number of observations for each regime, as this would affect the robustness of the estimates.

Therefore, considering the small sample size of only 49 observations at hand and the visual analysis

of the data, a single threshold is assumed to be present. The threshold will divide the data set into two

regimes, i.e., an upper and a lower regime. The upper regime corresponds to expansionary

macroeconomic environment when LSM growth is high. On the other hand, the lower regime

corresponds to tight macroeconomic environment when LSM growth is either very low or negative.

17

The selected threshold variable, i.e., the length of moving average and delay parameter are based on a

grid search methodology described below.

Four specifications of the threshold variable are considered. LSM growth rate with a delay of

1 and 2, and 2 period moving average of LSM growth with a delay of 1 and 2. The length of moving

average and the delay is not extended further keeping in view the limitations of the small dataset at

hand. Each threshold variable is then sorted in ascending order. 15 percent (each) of observations are

excluded from both ends of the sorted variable to have a buffer, such that the threshold value divides

the whole sample into two sufficient sized sub-samples. The threshold model of the form described

previously is estimated with each possible threshold value(𝛾), and for each estimated regression the

Root Mean Squared Error (RMSE) for the GNPLR equation of the Threshold VAR is determined.

Table 4 provides key statistics of the grid search for threshold from the four selected threshold

specifications.

As may be seen in Table 4, the threshold variable LSMG with moving average length of 2 and

delay length of 1 has the minimum RMSE among all four specifications. Further, the same

specification has the largest standard deviation, which implies high variation in RMSE as the threshold

variable crosses through the values, signifying the best threshold effect. The same specification is used

for the rest of the analysis.

Table 4. Grid Search for Threshold Specification

Grid Search Statistics (RMSE)

Threshold Variable Min Avg Max St. Dev.

LSMG d=1

0.00304 0.00353 0.00394 0.00028

LSMG d=2

0.00310 0.00354 0.00378 0.00018

LSMG MA(2), d=1

0.00293 0.00341 0.00393 0.00033

LSMG MA(2), d=2

0.00313 0.00341 0.00386 0.00024

Grid Search for Threshold Specification.

The scatter plot of GNPLR vs the selected threshold variable is given in Figure 3. Comparing

the linear and power fit lines, a clear non-linear relationship is found to be present.

18

Figure 3. Scatter Plot of Gross NPL Ratio against Threshold Variable

.06

.08

.10

.12

.14

.16

.18

-.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12 .14

2 Period Moving Average of LSMG(-1)

Gro

ss N

PL R

atio

Linear Fit

Power Fit

Figure 4 provides the result of grid search for the selected threshold variable, i.e., the plot of

RMSE of regression against the threshold variable. It is evident from Figure 4 that as the threshold

variable increases the RMSE initially declines, reaching a minimum around a value of 0.02 of the

threshold variable and then rises sharply above value of 0.0339 of the threshold variable, further

establishing evidence for presence of a threshold. Therefore, it is reasonable to assume that the

threshold value lies somewhere between the range of 0.02 to 0.034 of the threshold variable.

Figure 4. Regression RMSE vs. Threshold Variable

.0028

.0030

.0032

.0034

.0036

.0038

.0040

-.01 .00 .01 .02 .03 .04 .05 .06 .07 .08

2 Period Moving Average of LSMG(-1)

RMSE

Next, in the context of the subject vector model, to determine the threshold value

endogenously at which the threshold is found to be significant, a formal test is incorporated. The test

for significance of the threshold is to test whether the model parameters significantly differ across

19

both regimes. In this case, this is equivalent to testing H0: θ1 = θ2, against Ha: θ1 ≠ θ2. For each

possible threshold value, the Wald statistic testing the hypothesis of no difference between regimes

was calculated. However, standard distribution for Wald test is not applicable here since the null

hypothesis is that there is no threshold, and therefore, under the null, 𝛾 will be unidentified (similar

to the argument for the modified distribution for ADF test). Therefore, Hansen’s (1996) method of

simulation with 500 replications is used to generate the relevant distribution for hypothesis testing.

The value at which the log determinant of the residuals is maximized is said to be the estimated

threshold value.

The theoretical foundations of Balke (2000), and the codes provided therein, are utilized for

this study to formally construct the threshold VAR model, determine the threshold value, test its

significance, and generate the corresponding non-linear impulse response functions.

4.4 Forecasting Methodology

Since the primary objective is to be able to forecast the Gross NPL Ratio well in advance

based on the current shocks, therefore, to compare the models for forecast accuracy, both in sample

static and out of sample dynamic forecasts are constructed for Gross NPL Ratio. Dynamic forecasts

are constructed using the actual observed values of the lagged independent variables, and the lagged

forecasted value of the dependent variable for each subsequent forecast. This means that a forecast

for period t+2 using a single lag model will utilize the actual values of independent variables observed

in period t+1 and the forecasted value of dependent variable in t+1. Therefore, single equation

forecasts are particularly useful in stress testing using scenario analysis.

In sample forecasts are constructed by first estimating the models till 2016Q1 and then

forecasting from the start of the sample. For the evaluation of in sample forecasts, ‘Absolute

Percentage Error’ (APE) in forecasting is calculated for both models for each period. The average of

APE provides the in sample ‘Mean Absolute Percentage Error’ (MAPE) for each model. The use of

MAPE for comparing the models’ forecast accuracy instead of Root Mean Squared Errors is due to

the scale independence and generally easier interpretation associated with the former.

Since the primary objective is to forecast the future values based on past and current inputs to

the model, out of sample forecast accuracy must be evaluated. For this purpose, a different approach

is adopted. The sample is initially restricted to 2016Q1 and the models are estimated. Then a forecast

is constructed for a horizon of up to four period ahead i.e. from 2016Q2 to 2017Q1. Afterwards, in

each iteration, an observation is added to the sample recursively while keeping the start of the sample

20

fixed, and the forecast period is moved forward accordingly. Table 5 provides the details of estimation

and forecast periods for each forecast.

In essence, there will be five t+4, six t+3, seven t+2, and eight t+1 forecasts. After the

forecasts are constructed, Absolute Percentage Errors (APE) in forecasts are calculated similar to the

in sample forecasts. The errors are averaged for all n-step ahead forecasts. That is, errors of all t+1

forecasts are averaged to calculate MAPE for 1-period ahead forecasts. Likewise, MAPE is calculated

for 2-, 3-, and 4-period ahead forecasts as well. The 1-, 2-, 3-, and 4-period ahead MAPE is calculated

for both linear and threshold models and the results are compared to determine which model, on

average, makes the lowest error in forecasting GNPLR.

Table 5. Out of Sample Forecast Methodology

Estimation up to Forecast Horizon Forecast Period

2016 Q1 t+1 to t+4 2016 Q2 to 2017 Q1

2016 Q2 t+1 to t+4 2016 Q3 to 2017 Q2

2016 Q3 t+1 to t+4 2016 Q4 to 2017 Q3

2016 Q4 t+1 to t+4 2017 Q1 to 2017 Q4

2017 Q1 t+1 to t+4 2017 Q2 to 2018 Q1

2017 Q2 t+1 to t+3 2017 Q3 to 2018 Q1

2017 Q3 t+1 to t+2 2017 Q4 to 2018 Q1

2017 Q4 t+1 2018 Q1

Recursive Estimation and Forecast Horizon.

5. Results

The results of the test for significance of threshold, along with the sup-Wald, avg-Wald, and

exp-Wald statistics are provided in Table 6.

Table 6. Threshold Test for Non-Linearity

Wald Statistics

VAR in Threshold Variable Estimated

Threshold Value Sup-LM Avg-LM Exp-LM

Level LSMG MA(2), d = 1

ϒ = 0.033930 127.75 (0.000)

97.31 (0.000)

62.04 (0.000)

Threshold test for non-linearity, system includes Gross NPL Ratio, Interest Rate, Inflation, and LSM Growth.

21

The tests suggest presence of threshold effect in the model with high significance. It is found

that the estimated threshold value lies within the threshold region earlier identified in Figure 4, i.e.,

the plot of regression RMSE against the threshold variable.

Table 7 provides the Mean Absolute Percentage Error (MAPE) of the in sample static forecast

of Gross NPL Ratio. The MAPE is calculated for both models and the threshold value used to

calculate in sample forecast MAPE of the threshold model is based on the threshold test results. It

can be seen that the linear model has an in sample forecast MAPE of 2.57. Whereas, the threshold

model provides a better fit to the data and improves the in sample forecasts compared to the linear

counterpart, as the MAPE for threshold model is 1.93. It is evident that the Threshold VAR model

provides the lowest error and the best fit to the data.

Table 7. MAPE: In Sample Forecast

VAR

Linear Model Threshold Model

Threshold Value - ϒ = 0.03393

Mean Absolute Percentage Error (%) 2.57 1.93

Mean Absolute Percentage Error (MAPE) for in sample forecast: 2006:Q1 to 2016: Q1.

Figure 5 shows the run plots of the residuals of Gross NPL Ratio equation from both linear

and threshold VAR models. It can be seen that the threshold model has lower errors and deviations

compared to linear model.

Figure 5. Run Sequence Plots of Residuals of the Models

-.016

-.012

-.008

-.004

.000

.004

.008

.012

2006:Q1

2007:Q1

2008:Q1

2009:Q1

2010:Q1

2011:Q1

2012:Q1

2013:Q1

2014:Q1

2015:Q1

2016:Q1

Linear Model

Threshold Model

22

No matter how closely a model may fit the in sample data; in terms of forecasting, the out of

sample forecasts are necessary to gauge the accuracy of the model to predict the future values, which

are not known a priori. Accordingly, the out of sample forecasts are constructed for all models based

on the methodology provided in Table 5. For the threshold models, the forecasts are constructed for

all the potential threshold values identified through the RMSE plots. The Mean Absolute Percentage

Error (MAPE) for the out of sample dynamic forecasts of Gross NPL ratio through each model is

provided in Table 8.

It can be seen that the linear VAR model produces a MAPE ranging between 4.13 for t+1,

and 6.34 for t+4 forecasts, whereas the average MAPE in the full horizon context is 5.18.

Table 8. MAPE: Out of Sample Forecast

VAR in Levels

Threshold Value

(ϒ = )

Mean Absolute Percentage Error (%)

t+1 t+2 t+3 t+4 Full Horizon

Linear Model --- 4.13 4.42 5.83 6.34 5.18

Threshold Model

0.0200 5.32 6.76 5.11 5.53 5.68

0.0240 5.62 6.40 6.16 5.57 5.94

0.0290 5.71 7.30 7.01 6.87 6.72

0.0312 5.34 6.79 5.53 5.93 5.90

0.0313 4.99 5.17 3.46 4.00 4.41

0.0317 5.05 4.91 3.37 3.64 4.24

0.0339 4.07 3.43 2.42 2.99 3.23

0.0353 4.72 4.70 3.81 2.40 3.91 Mean Absolute Percentage Error (MAPE) for out of sample forecast (Recursive): Forecasts from 2016:Q2 to 2018: Q1.

Moreover, in case of threshold VAR model, it is interesting to note that as the threshold values

rise the forecast errors decline and reach the minimum at a threshold value of 0.0339. It is worth

noting that this value coincides with the threshold value earlier identified and estimated through the

threshold test reported in Table 6. The threshold VAR model with endogenously estimated threshold

value of 0.0339 provides the lowest error in forecasts particularly at longer horizon. The MAPE from

the threshold VAR is 4.07 for t+1, 3.43 for t+2, 2.42 for t+3, and 2.99 for t+4 forecasts, whereas for

the full horizon, MAPE is 3.23 which is much lower than the linear model. This hints at the usefulness

of the non-linear model for stress testing since the purpose therein is to predict well in advance the

future reactions to current hypothetical or actual shocks.

23

In order to further test the forecast accuracy of the threshold VAR model around the peak

historical values, the values of GNPLR from 2010Q3 to 2011Q4 (the period of sudden sharp rise and

peak) are removed from the sample and the model is estimated over the remainder sample (start to

end except the mentioned period) and the dynamic forecast for this period is then constructed. Using

the same methodology, the forecast is also constructed for the linear VAR model. The forecast results

are provided in Figure 6. It can be seen that the threshold model fares much better at predicting the

sharp rise and peak in Gross NPL Ratio compared to the linear model.

Figure 6. Out of Sample Forecast around the Historical Rise and Peak

.12

.13

.14

.15

.16

.17

2009:Q1 2010:Q1 2011:Q1 2012:Q1 2013:Q1 2014:Q1 2015:Q1

Gross NPL Ratio (Actual)

Forecast (Linear)

Forecast (Threshold)

One of the benefits of constructing a non-linear model is that the impulse responses generated

from it are not necessarily symmetric i.e. positive and negative shocks may yield different shapes of

responses, and also that small and large shocks may not necessarily be a multiple of each other i.e.

small and large shocks may also yield different shapes of responses. Threshold model is particularly

useful in the sense that two different sets of impulse responses are possible to be calculated: (i) for the

lower regime, and (ii) for the upper regime. These two sets of responses may not necessarily be similar

in shapes. The non-linear generalized impulse responses of Gross NPL ratio to positive and negative

1 and 2 standard deviation shocks to each variable in the system are constructed for both upper and

lower regimes. The results are provided in Figure 7 (panels a & b).

As may be seen in Figure 7, all responses generated from the threshold model are intuitively

correct and differences in small and large shocks and positive and negative shocks are clearly visible.

Also, the responses in the two regimes are quite different. These are compared to +1 SD shocks from

the linear VAR model given in Figure 8.

24

Figure 7. Threshold VAR Model: Generalized Impulse Responses of Gross NPL Ratio to +/- 1&2 SD Shocks to All Variables

(a) – Upper Regime Impulse Responses

-.008

-.006

-.004

-.002

.000

.002

.004

.006

.008

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to GNPLR

-.015

-.010

-.005

.000

.005

.010

.015

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to Inflation

-.0100

-.0075

-.0050

-.0025

.0000

.0025

.0050

.0075

.0100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to Interest Rate

-.006

-.004

-.002

.000

.002

.004

.006

.008

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to LSM Grow th

(b) – Lower Regime Impulse Responses

-.008

-.006

-.004

-.002

.000

.002

.004

.006

.008

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to GNPLR

-.0100

-.0075

-.0050

-.0025

.0000

.0025

.0050

.0075

.0100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to Inflation

-.015

-.010

-.005

.000

.005

.010

.015

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to Interest Rate

-.006

-.004

-.002

.000

.002

.004

.006

.008

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

+2SD -2SD

+1SD -1SD

Response of GNPLR to +/- 1 & 2 SD shocks to LSM Grow th

25

It can be seen that in both regimes, positive shocks to GNPLR have positive impact on

GNPLR, however the response is larger in the upper regime (easy macroeconomic environment) as

compared to the lower regime (tight macroeconomic environment). Moreover, positive shocks to

inflation and interest rates have positive impact on GNPLR in both regimes, however, the interest

rate shocks translate very slowly in the lower regime as compared to the upper regime. The response

of GNPLR to positive shocks to LSM growth is negative. One of the interesting findings of this study

is that positive and negative small shocks (1 SD) to LSM growth are somewhat symmetrical in both

regimes, however, large shocks (2 SD) are not symmetrical in the two regimes. The impulse responses

show that negative shocks to LSM growth have a more profound effect in the upper regime compared

to lower regime. On the other hand, positive shocks to LSM Growth have a more profound effect

when the economy is in the lower regime as compared to the upper regime.

Figure 8 provides the impulse responses generated from the linear VAR model. Since the

linear VAR model can only produce unit impulse responses that are insensitive to sign and size of

shocks, therefore it can be seen that the linear impulse responses significantly differ from the non-

linear model’s generalized impulse responses.

Figure 8. Linear VAR Model: Impulse Response of Gross NPL Ratio to +1 SD Shock to All Variables

-.008

-.004

.000

.004

.008

.012

1 2 3 4 5 6 7 8 9 10

Response of GNPLR to +1 SD Shock to GNPLR

-.008

-.004

.000

.004

.008

.012

1 2 3 4 5 6 7 8 9 10

Response of GNPLR to +1 SD Shock to Inflation

-.008

-.004

.000

.004

.008

.012

1 2 3 4 5 6 7 8 9 10

Response of GNPLR to +1 SD Shock to Interest Rate

-.008

-.004

.000

.004

.008

.012

1 2 3 4 5 6 7 8 9 10

+1 SD Shock ± 2 S.E.

Response of GNPLR to +1 SD Shock to LSM Growth

26

6. Conclusions

This study has attempted to develop a Threshold Vector Autoregressive model for the banking

sector of Pakistan and compared it to the conventional linear VAR model in an attempt to provide

more accurate forecasts of the Gross NPL Ratio of the aggregate banking sector; a key financial stress

indicator. The analysis has been carried out on a system of variables that included Gross Non

Performing Loans ratio, Inflation rate, SBP Reverse Repo rate, and LSM Growth rate. Graphical

representation of the data has been used to understand the patterns of non-linearity, and it is found

that LSM Growth rate has a nonlinear relationship with all the variables in the system. Since financial

stress does not emerge from a one period slowdown but a continued slowdown of the economy,

therefore it is assumed that the accurate variable as determinant of the threshold in the system is the

lagged 2 period moving average of the LSM Growth rate.

To gain further insight, the GNPLR equation of the non-linear system was estimated with all

the possible values of the threshold variable and the RMSE from each regression was plotted against

the values of the threshold variable. Presence of a threshold was established and further tests of non-

linearity were employed to test the significance of the threshold in the context of the vector model. A

highly significant threshold has been found i.e., when the average LSM Growth in the past two

quarters was 3.39%, thus dividing the system into an upper regime (easy macroeconomic environment)

and a lower regime (tight macroeconomic environment).

The model was then used to construct in sample and out of sample forecasts and was found

to predict Gross NPL ratio with much lower forecast errors compared to the linear VAR model,

particularly when the forecast horizon was beyond one period ahead. It also fared better at predicting

the historical rise and peak values of Gross NPL ratio. Generalized impulse response functions have

been constructed using the threshold VAR model, which were found to be dissimilar in both regimes

and are also sensitive to the size and sign of the shocks.

We conclude that on the basis of the accuracy of the threshold model’s forecasts compared to

the linear model and other specifications, it stands out as more appropriate model to be used for stress

testing, stability analysis, and formulation of credit policies. However, since the data used for the

methodology is of the overall banking sector, and therefore, it may be subject to aggregation bias as

the borrowers in different sectors of the economy do not react similarly to macroeconomic conditions.

We therefore recommend that the present study should serve as a basis for further analysis, which

may be carried out using more granular, sector specific data to identify possible thresholds for each

27

credit class, which may increase forecast accuracy and provide further insights. The model may also

be further developed to incorporate smooth transitions around the estimated threshold value.

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