The Tobit cointegrated vector autoregressive model: An application to the currency market

Journal Pre-proof The Tobit cointegrated vector autoregressive model: An application to the currency market Wojciech Grabowski, Aleksander Welfe PII:

S0264-9993(19)30137-3

DOI:

https://doi.org/10.1016/j.econmod.2019.10.008

Reference:

ECMODE 5031

To appear in:

Economic Modelling

Received Date: 26 January 2019 Revised Date:

5 October 2019

Accepted Date: 5 October 2019

Please cite this article as: Grabowski, W., Welfe, A., The Tobit cointegrated vector autoregressive model: An application to the currency market, Economic Modelling (2019), doi: https://doi.org/10.1016/ j.econmod.2019.10.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

The Tobit Cointegrated Vector Autoregressive Model: An Application to the Currency Market. Wojciech Grabowskia Aleksander Welfeb, corresponding author

a – Chair of Econometric Models and Forecasts, University of Lodz, 37/39 Rewolucji 1905 r., 90-214 Lodz, Poland; [email protected] b - Chair of Econometric Models and Forecasts, University of Lodz, 37/39 Rewolucji 1905 r., 90-214 Lodz, Poland; [email protected]

Abstract The Tobit cointegrated vector autoregressive model proposed in this study extends the existing methodology by allowing the censored variable to be nonstationary. The approach requires deriving the distribution of the cointegration rank test and simulating new critical values. The empirical application refers to the currency market. It has confirmed that the exchange rate is driven by four main forces: inflation, terms of trade, the perception of the country-specific risk, and the state of the currency market. Temporary disequilibria in the currency market arise not only from the “fundamental” factors, but also from the contagion effect.

Keywords: cointegration, contagion effect, exchange rate models, censored variables JEL: C32, C35

1

1. Introduction The determinants of the real exchange rate in developing and emerging countries are analysed in many empirical studies concentrating on the global financial crisis years and in the post-crisis period (see e.g. Barbosa et al. 2018; Caputo, 2018). The awareness of the nonstationarity problem has also led some researchers to exploring the issue of instability in the currency markets (Balaga, Padhi, 2018). Nevertheless, two crucial aspects have so far been ignored. Firstly, because some market pressures do not lead to exchange rate movements, the variable measuring market disequilibrium caused by fundamentals must be modelled together with the exchange rate. Secondly, it is very natural that market pressure occurs in some periods of the sample but does not occur in others. It is for this fact that only the Tobit-type model seems to be an appropriate tool of analysis. In the light of numerous recent studies it has become clear that many phenomena cannot be adequately explained unless the censored variables are used: the Tobit-type models have been employed to identify the determinants of central banks’ interventions (see Ordonez-Callamand et al. 2018) and to explain stock return volatilities as well as jumps in equilibrium prices in foreign exchange markets (see El Ouadghiri, Uctum, 2016) or in female labour force supply (see Ye, Xu and Wu 2018). On the other hand, the existing research has provided solid evidence that most macroeconomic time series are generated by nonstationary processes (the majority of which are integrated of order one). Therefore the Tobit cointegrated vector autoregressive model (Tobit-CVAR) is developed in the paper to handle the nonstationarity problem of endogenous censored variables. With this approach, the distribution of the cointegration rank test must be derived and its critical values have to 2

be simulated. Building on the traditional concepts, we propose a new exchange rate model allowing for currency market pressures (temporary disequilibria) measured by variables constructed like those used in the early warning systems, the parameters of which are estimated within the Tobit-CVAR framework. This paper makes the following contributions to the economic literature. Firstly, the nonstationarity problem of endogenous censored variables is solved using an innovative approach based on the Tobit-CVAR model and a method for its estimation is proposed. Secondly, new critical values enabling inference about the cointegrating rank in the case of a presence of censored integrated of order 1 variable are derived. Thirdly, the existing approach to real exchange rate modelling is extended and new disequilibrium indicator is constructed. It is probably the first attempt to model the exchange rate allowing for the endogeneity of the censored variable. The structure of the paper is the following. In section two, the Tobit-CVAR model estimation method is provided. In section three, critical values are derived to enable inference about the cointegration rank. The empirical results obtained with the currency market model are presented in section four. The last section concludes.

2. The Tobit-CVAR model – estimation and testing The qualitative vector autoregressive model, Qual-VAR (see Dueker, 2005), which is the multivariate extension of the dynamic probit model (see Eichengreen, Watson, Grossman, 1985), transforms into the Tobit-VAR:

Π(L )y t = µ + ε t ,

(1) 3

if the qualitative variable is defined as yM +1,t

 yM* +1,t if yM* +1,t > 0 = otherwise 0

(2)

yt   ~ and vector y t =  *  consists of M observable variables included in ~ y t and the  y M +1,t 

censored variable yM* +1,t ; Π (L ) is a set of (M + 1) × (M + 1) matrices, for L=0,…,S with the identity matrix at L = 0 ; µ is a vector of intercepts. The error term ε t is expected to have a multivariate normal distribution N(0, Ω ⊗ I ) . The Tobit cointegrated vector autoregressive

representation, Tobit-CVAR, of the model is given by: S −1

∆y t = AB T y t −1 + ∑ γ s ∆y t − s + ε t ,

(3)

s =1

under the assumptions that y t consists of I(1) variables and matrix Π = AB T is of reduced rank ( B and A have standard interpretations of cointegrating vectors and weights). In order to estimate the system of equations (3), the cointegration rank must be determined (see section 3). The procedure is conditional on the size of cointegration space and consists of the following steps. Firstly, the initial values A (0 ) , B (0 ) , Ω (0 ) and γ (s0 ) for s=1,2,…,S-1 are set at the point estimates of the CVAR. The censored variable is treated as

continuous. Secondly, using the Geweke-Hajivassiliou-Keane (GHK) smooth recursive conditioning simulator the theoretical values of the censored variable are calculated (see Borsch-Supan, Hajivassiliou, 1993; Keane, 1994). Thirdly, the parameters included in A (1) , B (1) , Ω (1) and γ (s1) are estimated by the reduced rank regression method (see Johansen,

4

1991, 1996). The estimates thus obtained are the initial values of the next iteration. The procedure is continued until convergence is achieved. The TRACE statistic that is commonly used to test the rank of Π is TRACE = −T

∑ ln(1 − λˆ ), M

(4)

i

i = r +1

where r denotes the size of the cointegration space and eigenvalues λˆ1 , λˆ2 ,K, λˆM are the solutions of the equation: −1 λS11 − S10 S 00 S 01 = 0 ,

(5)

T T T ′ where S11 = T −1 ∑ y tT−1y t −1 , S 00 = T −1 ∑ ∆y tT−1∆y t −1 , S 01 = T −1 ∑ ∆y tT−1y t −1 , S10 = (S 01 ) . t =1

t =1

t =1

The asymptotic distribution of the TRACE statistic in the presence of the censored variable is given by (see Appendix 1)

(

d TRACE  → tr [A ] B Ω − 1B T −1

)

(6)

where 1  2  ∫0 W(r ) dr A = 1   ∫ W (r )1V > 0 [V (r )]dr  0

 1  ∫ dW(r )W(r )dr 0 B = 1   ∫ W(r )1V >0 [dV (r )]dr  0



1

∫ W(r )1 [V (r )]dr  V >0

0

   ∫ 1V > 0 [V (r )]dr    0  1

2

,   



1

∫ W(r )1 [dV (r )]dr  V >0

0

1   ∫ 1V >0 [dV (r )]dr    0 

2

,   

W (r ) is a multivariate Wiener process and V (r ) is a univariate Wiener process.

5

Because in the case of censoring, the asymptotic distribution of the trace statistic differs from the standard one and depends on the percentage of censored observations, new critical values have to be simulated. They are presented in Appendix 2.

3. Application of the Tobit-CVAR to the exchange rate modelling The literature offers numerous theoretical explanations of why currency crises occur. The so-called first generation models hold that the timing of speculative attacks is the result of inconsistencies between government policies and a fixed exchange rate (Licchetta, 2011). Consequently, they concentrate on the relationship between the probability of a currency crisis and fundamental variables such as current account, fiscal deficits, credit growth, and the measures of economic activity (see Krugman, 1979, Obstfeld, 1996, Broner, 2008, Sevim et al. 2014). The second generation models indicate the role of self-fulfilling expectations as factors leading to currency crises when the strategic complementarities between agents result in multiple equilibrium. Sachs et al. (1996) argue that the macroeconomic fundamentals are important too, as they significantly determine the range of likely equilibria. The failure of both classes of models to explain the currency crisis in South-East Asia in 1997-1998 led to the emergence of a third generation of models, which concentrate on the degree of liquidity of the banking system, the existence, or not, of government guarantees, and the role of currency mismatches in creditconstrained economies. Within their framework, a substantial amount of short-term external liabilities, weak banking supervision, and a high ratio of non-performing loans-to-gross loans are indicated as increasing the risk of a crisis. Moreover, the third-generation models 6

consider also the transmission of negative shocks from the stock market and a contagion effect between neighboring economies or markets (see Candelon et al. 2014; Grabowski, Welfe, 2016; Jin, An, 2016; Leung, Schiereck, Schroeder, 2017; Roy, Roy, 2017). The contagion effect driven by the herd behavior of mainly stock market investors has two major sources (see Wermers, 1999, Bohl, Brzeszczyński, Wilfling, 2009). The first of them is the diversification of portfolios (see: Ait-Sahalia, Hurd, 2016): as the number of portfolio assets increases, seeking information about the situation of particular countries in a region becomes uneconomic (see Calvo, Mendoza, 2000). The second one is investors having different access to information (asymmetry), which causes that the informationally disadvantaged ones seek to follow those perceived as more knowledgeable (see Froot et al., 1992) or better informed (a so-called “bandwagon effect”, see Calvo, 1998). As a result, some investors may act on information that has little to do with fundamentals, a tendency amplified by the awareness that the penalty for missing a bull market is (usually) severe, whereas losses related to the bear market are more tolerable as most investors are losing then (see Kumar, Persaud, 2002). In the case of the currency market, the same effect may come from multiple equilibria and self-fulfilling prophecies (see Jeanne, 1997; Beckman, Czudaj, 2017). The signals about a deteriorating macroeconomic condition of a country that investors receive may trigger simultaneous and mass outflow of capital from this country, as well as from other countries in the region. At the same time the bandwagon effect can create a sort of “hot money” in foreign exchange markets, which sometimes makes them overreact to the news about the prospects of national economies. The herd effect means that the sudden 7

depreciation of one country’s currency causes the currency of another country in the region to follow suit. Moreover, a stock and currency markets are reported to be significantly related to each other (see for example Khalid, Kawai, 2003, Tian, Ma, 2010, BahmaniOskooee, Saha, 2015, 2016). Therefore, an increasing spread between the indices of the domestic and foreign stock markets is likely to reduce the risk of currency market instability what should be taken into account in the analysis (the results of some studies point to significant causality; see Bahmani-Oskooee, Saha, 2015). Summing up, the interaction between the state of the currency market and other macroeconomic variables can be written as the following, long-run relationship:

(

)

(

)

N t* + β 1 id t − id t f + β 2 CTGt + β 3 NFAt + β 4 BC t + β 5 Dt − Dt f = ε 1t ,

(7)

where N t* denotes the (unobserved) state of the currency market; idt is the (main) stock market index; CTGt measures the contagion effect; NFAt is the net foreign assets-to-GDP ratio; BCt is the level of business confidence; Dt is a debt-to-GDP ratio; ε 1t represents white noise; the superscript f indicates the reference country(s); and the small letters stand for natural logarithms. The purchasing power parity hypothesis (PPP) that assumes the exchange rate to be driven by relative prices between countries is probably the first theory to explain the exchange rate dynamics (see e.g. MacDonald 2007). The assumption that prices are determined by domestic nominal money supply and real money demand result in a model where the exchange rate depends on money supply disparity, interest rate and income differentials. However, ample evidence of persistent deviations from a PPP path gave an impulse for further studies (see e.g. Bergstrand, 1991). 8

The observation that international trade is unable to equalize the prices of nontradeable goods paved the way for the Balassa-Samuelson effect hypothesis (see Balassa 1964, Samuelson 1964) explaining why the exchange rate is higher in more productive countries (see e.g. Canzoneri et al. 1999; Cheung et al. 2017). The Balassa-Samuelson effect may be relevant in the case of developing economies, particularly in the early stages of the catching-up process (see Konopczak and Welfe, 2017). More recently, a surge in the interest in the stock-flow approach to modelling exchange rate fluctuations has been noted (see Alberola et al. 2002). According to this approach, the equilibrium real exchange rate depends on the stock and flows of assets between countries, so a rise in net foreign liabilities is likely to result in real exchange rate depreciation. In the post-Keynesian exchange rate models, aggregate expectations are assumed to have an important effect on the exchange rate path. Hence, the best strategy for currency dealers and fund managers is to anticipate the expectations of market agents which are in many applications approximated by actual level of portfolio (see Harvey, 2009; Kaltenbrunner, 2015; Barbosa et al. 2018). The variety of approaches to modelling the equilibrium exchange rate can be easily sorted out based on their time horizons (see the survey in MacDonald 2007). The PPP well explains the behaviour of the exchange rate over very long horizons. For shorter periods, the behavioural equilibrium exchange rate approach (BEER; MacDonald 2007) and the capital enhanced equilibrium exchange rate approach (CHEER; see Johansen, Juselius, 1992; Juselius, MacDonald, 2004; Kębłowski, Welfe, 2010) seem more appropriate. Both 9

BEER and CHEER use the balance of payments equilibrium condition, but BEER encompasses CHEER when the debt liquidity index is substituted by the difference in credit default risk premiums. The recent financial crisis has shown that two more factors must be taken into account. The first of them is tensions in the currency market of a country caused by economic and political instability. As they provoke investors to sell securities denominated in the country’s currency and repatriate funds, the downward pressure on the currency increases (Hui, Chung, 2011). The second factor is investment risk that investors have become increasingly aware of in the last two decades, and which is typically understood as the difference between the credit default swaps indices (CDS) in the country under consideration and the reference country (see Kębłowski, Welfe, 2012, Aizenman et al., 2013, Akdogan, Chadwick, 2013). The above hypotheses result in the following long-run relationship: ext − ( pt − ptf ) + θ1 (CDS t − CDS t f ) + θ 2 N t* + θ 3tot t = ε 2t ,

(8)

where ext denotes the spot exchange rate, p t is the price index, CDSt stands for credit default swaps, tott denotes terms of trade, ε 2t represents white noise. Directly unobservable, the state of the currency market is represented by variable N t* and explained by equation (7). According to the monetary approach prices are determined by domestic nominal money supply and real money demand. Therefore the following relation is considered as well:

(

)

(

)

(

)

( pt − ptf ) + κ 1 m2 t − m2 tf + κ 2 I t − I t f + κ 3 xt − xtf = ε 3t , 10

(9)

where m2t denotes money supply M2, I t is the market interest rate, and xt denotes the level of output, ε 3t represents white noise. It is important to note in conclusion that with a system of three cointegrating vectors the spot exchange rate is guaranteed to depend on ( pt − ptf ) which is explained by the monetary factors (equation (9)), and on the market pressure, N t* , determined by variables postulated by the stock-flow and post-Keynesian theories (equation (7)). All these factors indirectly affect the exchange rate in the long-run. The empirical investigation of the whole model is restricted by the limited availability data for N t* . The first proposals involving the construction of a market pressure indicator built on the observation that the monetary reserves and the interest rates are economic measures that the authorities in countries with the floating exchange rate regime can use to control the market (Eichengreen, Rose, Wyplosz, 1995):

 EX t − EX t −1   RS − RSt −1   − w2  t  + w3 I t − I t f − I t −1 − I t f−1 , EMPt = w1  EX RS t −1 t −1    

((

) (

))

(10)

where EMPt denotes the market pressure, EX t – the exchange rate, RSt – the level of currency reserves to monetary base ratio; I t – the market interest rate, w1 , w2 and w3 are proportional to the inverse standard deviations of the respective variables in the sample and sum up to 1 (for some modifications see Klaassen, Jager, 2011). However, the above measure based on the growth rates is only capable of identifying the short-term pressures. Grabowski and Welfe (2016) proposition is free of this deficiency:

11

~  EX t − EX t EMS t = w1  ~ EX t 

~   RS t − R S t  − w2  ~   R St  

((

) (

))

  + w3 I t − I t f − I~t − I~t f ,  

(11)

n n ~ ~ n ~ where EMSt denotes the market state, EX t = ∑ b j EX t − j , I t = ∑ b j I t − j , I t f = ∑ b j I t f− j . The j =1

j =1

j =1

~ ~ ~ values of EX t and I t − I t f can be respectively interpreted as the exchange rate and interest rate disparity levels that the economic agents have learnt to accept. Because the currency reserves are realizations of stationary or trend-stationary stochastic processes (due to their nature confirmed by testing), it is quite inappropriate to assume that they may have any

~ “normal” level. Therefore, RSt can be determined by filtration (the simplest solution is the Hodrick-Prescott filter). The weights of b j that sum up to one decrease as the lags increase, which is rather self-explanatory:

bj =

bj =

1  n n  n n  1   , for an even n , +   + 1 − j 1 j ≤  +  − j 1 j >   2 n  2 2  2 2  n   +1 2 1  n +1  1 + − j 2 , for an odd n . n  2  n +1 2

(12a)

(12b)

Because the memory of economic agents is quite short, n of 6 seems appropriate for studies based on the monthly data what is confirmed by our results. From (11) it follows that the EMS increases as the exchange rate or interest rate disparity starts to grow beyond an acceptable level, or when the currency reserves fall below a level determined by the long-term trend. From the foreign investor or monetary 12

authorities’ perspective, the low and negative values of the EMS indicate that the situation in the currency market is stable. It is irrelevant if the situation is “very stable” or “only stable”. However, an EMS value exceeding the sample mean by one standard deviation shows that the currency market situation is likely to be unstable. It can also be assumed that if EMS exceeds the sample mean by two standard deviations, the problem of instability into the currency market turns serious. This observation leads to the following construction of a variable for measuring the depth of market instability:

N t = max(0, EMS t − µˆ (EMS ) − σˆ (EMS ))

(13)

where µˆ and σˆ are the sample mean and the standard deviation of the EMSt , respectively. We advocate to use measure (13) instead of (11) because the investors and policy makers usually want to know, not only if there is a problem of instability in the currency market, but how severe it is. We are aware of the fact that variable EMSt is stationary (value of the ADF statistic used for testing stationary equals -7.59 and p-value equals 0.00). However it has been observed that in the periods of speculative bubbles in the financial markets the rates of return are nonstationary (see for example Homm, Breitung, 2012, Harvey et al., 2016). This explains why the unobservable latent variable (related to the variable defined by formula (11)) is expected to be integrated of order one, which means that instability in the currency market can last longer. The substitution of Nt defined as above for the unobservable variable N t* allows the application of the Tobit-CVAR model and an empirical analysis of the exchange rate behavior in the presence of market pressures. 13

4. Empirical results

Poland has been selected for empirical analysis for at least three reasons. Firstly, Poland, a medium-sized country, belongs to the group of New EU Member States where a strong contagion effect can be expected. Secondly, the Polish zloty exchange rate (PLN/EUR) is floating and fully determined by the market forces. Thirdly, the convergence process and profound economic changes in Poland occasionally create tensions in the currency market. This description seems to be relevant also to other countries with similar characteristics. The monthly EUROSTAT, IMF and Polish Central Statistical Office data used in the study span the period from January 2001 to December 2018. As the base year for constant prices the year 2005 was adopted. Since Germany accounts for over 40 percent of Polish foreign trade, it was selected as the reference country (see formula (7) and following). Poland, the Czech Republic and Hungary lie in the same geographical region, share the history of transition from centrally-planned economies to a market system after 1990, and have recently joined the European Union. Because of these similarities, it was assumed that the Polish currency market is strongly affected by developments in the other two countries. In the past, negative sentiments in the global financial markets frequently contributed to simultaneous outflows of foreign capital from the whole region. A confirmation of this mechanism can be found in the studies investigating the exchange rate behavior (Kębłowski, 2011, Bubak et al., 2011), responses to the monetary policy shocks 14

(Jarociński, 2010), and interactions between financial markets (for example Yang, Hsiao, Li, Wang, 2006 Hanousek, Kocenda, Kutan 2009, Bieńkowski, Gawrońska-Nowak, Grabowski, 2014). Therefore variable CTGt was defined as follows:

{{

}{

}}

CTGt = max I N tCZ > 0 , I N tHU > 0 ,

(14)

where the upper indices CZ and HU refer to the Czech Republic and Hungary, respectively. To calculate N tCZ and N tHU , formulas (10) and (12) were used. The results proved that most of variables in the model are integrated of order 1 (table 1). A few of them turned out to be stationary. Since the highest order of integration equals 1, and there are at least two variables integrated of order 1, cointegration may occur. On the basis of Akaike’s information criterion, a lag order of 4 was selected (results upon request). The system was found to have three cointegrating vectors (table 2). [Table 1] & [Table 2] The exogeneity of variables was tested sequentially: at first variable BCt turned out to be weakly exogenous (LR = 4.98; p-value=0.17), then CDS t − CDS t f was assumed to be weakly exogenous, which was also confirmed by the LR test (LR = 1.13; p-value=0.77). Then NFAt and Dt − Dt f were assumed to be weakly exogenous, which was confirmed by the LR test (respectively LR=2.35; p-value=0.50, and LR=3.11; p-value=0.37) The data does not allow to reject the remaining restrictions (exclusion, structural, normalizing and homogeneity) at p=0.05 ( LR = 16.123 ) allowing for economic interpretation of estimation results.

15

The Shapiro-Wilk (superior for small samples, see Yap, Sim, 2011), the Lagrange multiplier, and the White tests for residuals orthogonalised by the Cholesky decomposition confirmed that the error terms are normally distributed, homoscedastic and not autocorrelated (table 3). [Table 3] The three identified long-run equilibrium relationships representing, respectively, currency market instability, exchange rate and the domestic-to-foreign price are as follows (the t-Statistics are given under the parameter estimates):

(

)

(

)

N t* = 0.362+ 0.005 Dt − Dt f + 0.025CTGt − 0.116 id t − id t f − 0.002 BCt − 1.006 NFAt , (15.a) (3.12)

(4.49 )

(

)

(1.96)

(−2.34)

(

)

( −2.20)

(−2.51)

ext = pt − ptf + 0.640+ 0.333 N t* + 0.018 CDS t − CDS t f − 0.182 tot t ,

(15.b)

(p

(15.c)

t

)

(1.10 )

( 2.85 )

( 2.23 )

(

)

(

)

( − 2.44 )

(

)

− p tf = 0.829+ 0.007 m2 t − m2 tf + 0.008 I t − I t f − 0.149 xt − xtf , (10.79 )

( 2.13 )

(9.25 )

( − 23.26 )

and the matrix of adjustment coefficients equals − 0.089 − 0.032   0.121 A= − 0.056  − 0.021   0.037

0.053 − 0.082 − 0.226 − 0.027 − 0.015 0.041

− 0.032 − 0.002 − 0.145  , − 0.022 − 0.003  − 0.016

(15.d)

where the vector of endogenous variables for the conditional model ((see Ericsson, 1995;

[

Harbo et al., 1998) is y t = N t*

ext

p t − p tf

xt − xtf

id t − id t f

m 2 t − m 2 tf

]

T

.

The above results confirm that all three variables adjust to their own log-run trajectories (cointegrating vectors), as is usually expected. The adjustment parameter of 0.082 indicates that the exchange rate takes about 12 months to return to its long-term path 16

and that market instability continues after a shock for 11 months (1/0.089). The adjustment parameter of -0.145 shows that the relative price level is back on the long-term path after around 7 months. As instability in the currency market increases, currency depreciates via the adjustment mechanisms, which is reflected by the estimate 0.053. The parameter estimate for the second cointegrating relation in the equation explaining ∆N t* is negative and significant, meaning that if in the previous period the real exchange rate was below its long-run equilibrium level, the probability of currency market instability increases. Consequently, a central bank’s intervention meant to achieve exchange rate appreciation may have a negative impact on the currency market in the future. Moreover, the obtained result is in line with the findings reported by authors such as Caputo (2018), according to which sustained REER overvaluation can lead, after a time, to persistent instability in the currency market. The cointegrating vector estimation results are as expected. The deteriorating performance of the Polish stock market (vis-à-vis the German stock market) increases the probability of occurrence and the intensity of instability in the currency market. This relationship is in line with the asset market approach (Maitra, Varun, 2019) according to which the exchange rate is determined by capital flows and not trade flows. An increase in the stock prices causes an increase in the demand for domestic assets, resulting in the appreciation of the domestic currency, but falling stock prices cause exchange rates to depreciate, increasing the probability of currency market instability occurring. Further, tensions in the Czech or Hungarian currency markets increase the probability of the Polish market also being affected by instability (see the first cointegrating 17

vector). It turns out that the global nature of currency market linkages facilitate the transmission of disequilibria across currency markets within the same region (see e.g. Kadlackova and Komarek (2017)). This explains why a contagion effect triggered by investors’ tendency to perceive all countries making up a region as a homogenous group is a major factor in market instability. Consistent with previous studies, the probability of a crisis occurring has been found to increase with the government debt-to-GDP ratio (see e.g. Licchetta, 2011) because the Central Bank converts the domestic currency denominated debt into bonds denominated in foreign currency. This conversion depletes foreign exchange reserves and increases the probability of currency market instability. This finding can also be explained by the link between currency crises and debt crises (see Dreher, Herz, Karb, 2009). The level of business confidence has also been proven to considerably increase the probability of currency market instability, meaning that trust in a currency is largely determined by the predicted developments in the economy. When economic prospects are gloomy, investors tend to withdraw their assets thus increasing the probability of instability in the currency market. This mechanism has been empirically confirmed for the euro area sovereign debt crisis. There are also reports indicating that low business confidence is a factor contributing to financial crises (see Xie, Chen 2019). The strength of a currency is founded on the continuous accumulation of net foreign assets, which is even capable of offsetting downward pressures from deteriorating terms of trade and slower relative productivity growth (Caputo 2018). The positive association between the net foreign assets-to-GDP ratio and the currency market stability established in 18

this study is supported by the results of numerous studies on developed and emerging economies (see e.g. Chia et al. 2014). The long-run equation explaining the exchange rate is consistent with the monetary approach (see third cointegrating equation). An increase in the money supply at home leads to an equiproportionate depreciation (MacDonald 2007). Because an increase in real domestic income stimulates the demand for real balances and consequently reduces domestic prices, it results in offsetting exchange appreciation. By contrast, an increase in interest rate disparity decreases the demand for real balances, increases prices, and leads to an exchange depreciation. The effect of the terms of trade on the exchange rate is negative (see equation (15b)). This finding is supported by an important strand of the literature where changes in the relative price of exports to imports are indicated as an important determinant of exchange rate movements (see e.g. Marion, 1994; Caputo, 2018). A positive effect of the difference between the credit default swaps for Poland and Germany on the level of the exchange rate has been shown. The result is unsurprising because market risk, as part of financial risk, plays an important role in determining exchange rate fluctuations in the catching-up economies with a floating exchange rate regime (see e.g. Caputo 2018). The existence of a relationship between the exchange rate and CDS spreads is supported by financial theory stating that the price of a financial asset depends on its riskiness (see e.g. Kębłowski and Welfe (2012)) and that an increase in country risk leads to the depreciation of the national currency. Lastly, currency market instability has been found to have a positive influence on the exchange rate level: mounting 19

currency market tensions cause the depreciation of the Polish currency against the euro in the long-run. In order to find out whether the Tobit-CVAR model better predicts the probability of currency market instability than the standard Tobit model, the models were compared for goodness of fit: ME1 =

1 T ( I {N t > 0}* P(N t* ≥ 0 ) + I {N t = 0}* P (N t* < 0)) , ∑ T t =1

(16.a)

ME 2 =

1 T ( I {N t > 0}* I {P (N t* ≥ 0) ≥ c}+ I {N t = 0}* I {P (N t* ≥ 0) < c}) , ∑ T t =1

(16.b)

where

is the fraction of ones in the sample. The ex-post forecast errors of the nominal exchange rate were compared for the full

system and the system without the exchange rate instability equation, respectively. Table 4 shows the goodness of fit for both cases, as well as the ex-post forecast errors. It is easy to see that the Tobit-CVAR model better predicts the probability and magnitude of tensions in the currency market than the standard Tobit model that lacks long-run equations explaining the exchange rate and the price ratio. Moreover, allowing for the presence of currency market instability and its determinants in the model improves the prediction of the longterm exchange rate. [Table 4] 5. Conclusions The inclusion of the censored variable in the CVAR model modifies the asymptotic distribution of the TRACE statistic that enables inference on the cointegration rank. The simulated values differ significantly from the standard ones if the percentage of censored 20

observations is high. The Tobit-CVAR model has been proven to be a more accurate forecasting tool compared with the standard Tobit model. The overarching conclusion from the empirical analysis is the following. Real exchange rate fluctuations are directly induced by market pressure (disequilibrium), a country risk as perceived by investors (measured by the relative CDS), and a country’s terms of trade. With regard to the state of the currency market, it is determined by countryspecific fundamentals and a contagion effect driven by investors’ propensity to perceive countries in the same group (or region) as homogenous. An approach to exchange rate modelling that separates forces affecting the spot exchange rate through the state of the currency market from those that have a direct effect seems to be especially useful for studying turbulent periods. The policy recommendations from the study are quite straightforward. Firstly, the impact of the variable associated with the contagion effect implies that central banks and other institutions in charge of macroeconomic policy should examine the likely consequences of instabilities (political, etc.) in other countries. Well-targeted preventive measures should be developed and employed early enough to make sure that they effectively protect the country from external shocks. Secondly, the estimates of the cointegrating vectors confirm that fiscal and economic policies have a major role in preventing currency market instability and sustained depreciation. The authorities should abstain from increasing the debt-to-GDP ratio. Thirdly, the focus of an economic policy should be on attracting foreign assets and boosting the confidence of business because both the factors decrease the probability of currency market instability occurring. 21

Acknowledgements We gratefully acknowledge the financial support from NCN MAESTRO 4/HS4/00612 and insightful referees’ comments. Any errors are the sole responsibility of the authors.

22

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Appendix 1. Asymptotic distribution of the TRACE statistic Assumption

1.

y t = y t −1 + w t ,

Let

∞

∞

w t = Π (L )e t = ∑ Π i e t −i ,

vt = ∑τ i ξ t −i with

i =0

∞

∑ iτ i =0

i

y0 = 0

i =0

y M* +1,t = y M* +1,t −1 + vt ,

and

Π (1)

∞

nonsingular,

∑i Π i =0

< ∞ . The innovations e t and ξ t are iid with mean zero and E ekt

elements of vector e t and E ξ t

r

r

<∞

i

where and

< ∞ for all

< ∞ for some r > 8 . These innovations have an

absolutely continuous distribution with respect to the Lebesgue measure and a characteristic function ϕ (t ) satisfying the condition lim t ϕ (t ) = 0 for some κ > 0 . κ

t →∞

LEMMA 1. If Assumption 1 holds, then there exists a probability space

(Ω, F , P )

supporting the sequences of random variables Vnt and Wnt satisfying the following:

(Vnt , Wnt ) = d

 1   T

t

∑v , i =1

i

1

t

∑w T i =1

i

 . 

(A.1)

If T t t − 1 WT (r ) = ∑ WTt I  ≤ r < , T  T t =1

(A.2)

and T t t − 1 VT (r ) = ∑ VTt I  ≤ r < , T  T t =1

(A.3)

29

a. s. a. s. then VT → V and WT → W in D[0,1] , the m-fold Cartesian product of the space

m

D[0,1] endowed with the uniform topology, where V and W are Brownian motions in

(Ω, F , P ) with standard deviation σ V

variance matrix Σ W .

In order to determine the asymptotic distribution of the TRACE statistic, the limit distribution of matrices S 11 , S 00 , S 01 must be derived. Distribution of the element (1,1) of matrix S 11 is obtained on the basis of standard theory for asymptotic distributions of integrated time series (Phillips, 1986): T

−1

∑y t =1

T

1

T

T ( m )t −1

y (m )t −1  → ∫ W(r ) W(r )dr . d

(A.4)

0

Distributions for elements (1,2), (2,1) and (2,2) follow from the asymptotic theory for nonlinear transformations of integrated time series (Park, Phillips, 1999): T

−1

∑ y( t =1

T

−1

T

m )t −1

y M +1,t −1  → ∫ W(r )1V >0 [V (r )]dr , d

(A.5)

0

1

T

∑y t =1

−1

1

T

M +1,t −1

y

T ( m )t −1

 → ∫ W(r )1V >0 [V (r )]dr , d

(A.6)

0

∑ (y T

t =1

)

2

M +1,t −1

2

1   →  ∫ 1V >0 [V (r )]dr  , 0  d

(A.7)

Matrix S 01 is decomposed analogously T

S 01 = T −1 ∑ ∆y Tt−1y t −1 = t =1

 T  ∑ ∆y (m )t −1y (m )t −1 =  Tt =1  ∆y M +1, t −1y ( m )t −1 ∑ t =1 T

.  t =1  T ∆yM +1,t −1 yM +1,t −1  ∑  t =1 T

∑ ∆y (

T m )t −1 M +1, t −1

y

30

(A.8)

Element (1,1) of matrix S 01 follows

T

−1

∑ y( t =1

T

1

T

m )t −1

∆y (m )t −1 → ∫ dV (r ) V (r )dr d

(A.9)

0

as Phillips (1986) showed. The limit distributions for consecutive elements of this matrix are based on the fact that they are recorded only for uncensored observations:

T

−1

∑ (∆y T

t =1

2

)

2

M +1,t −1

1   → ∫ 1V >0 [dV (r )]dr  , 0  d

T

1

t =1

0

(A.10)

d T −1 ∑ ∆y (m )t −1 y M +1,t −1  → ∫ W(r )1V >0 [dV (r )]dr ,

T

−1

1

T

∑y( t =1

(A.11)

m )t −1

∆y M +1,t −1  → ∫ W(r )1V >0 [dV (r )]dr . d

(A.12)

0

Finally, the limit distribution for the trace statistic is given by

(

)

d TRACE → tr [A ] B Ω − 1 B T , −1

(A.13)

where 1  2  ∫0 W(r ) dr A = 1   ∫ W(r )1V >0 [V (r )]dr  0

 1  ∫ dW(r )W(r )dr 0 B = 1   ∫ W(r )1V >0 [dV (r )]dr  0



1

∫ W(r )1 [V (r )]dr  V >0

0

1   ∫ 1V >0 [V (r )]dr    0 

2

,   

 ( ) [ ( ) ] W r 1 dV r dr  V > 0 ∫0 . 2 1     ∫ 1V >0 [dV (r )]dr      0   1

31

The above result finishes the proof.

Appendix 2. Simulation of critical values To simulate the critical values of the TRACE statistic, the following DGP was considered (see Osterwald-Lenum, 1992, Johansen, 1991, 1996 and MacKinnon, Haug, Michelis, 1999 for the case without a latent variable):

[∆~y

t

∆y Tt

]

T

= εt

(A.14)

where M − r denotes the number of independent common stochastic trends, εt are mutually independent, and ε t ~ N M −r (0, I ) . Without any loss of generality, the starting values of ~ y 0 and

y 0 can be set to 0. The percentage of censored observations, q, is assumed and the quantile of order q for variable ~ y t (denoted as ~ y q ) is calculated. The censored variable is generated in the following way:

z t = max(0, ~ yt − ~y q ) .

(A.15)

In the next step, the limit for statistic (4) is approximated by −1  T  T  T tr ∑ ε t S Tt ∑ S t S Tt  ∑ S t ε Tt  .  t =1  t =1  t =1 

(A.16)

In the course of this research five model structures were considered (this approach is consistent with Johansen’s (1996), but somewhat different from what Pesaran et al. (2000) advocated (see discussion in Turner, 2009):

[

* 1. S t = yˆ M +1,t −1

[

2. S t = yˆ M +1,t −1 *

T

]

y (m )t −1 , neither the constant nor the linear trend is present in the model,

]

y (m )t −1 1 , the constant is restricted to the cointegration space,

32

 

3. S t = z t −1 − z M − r −1

 

4. S t =  z t −1 − z M − r

t−

t−

1 (T + 1) , the constant is unrestricted, 2 

1 (T + 1) , the constant is unrestricted and the trend is restricted to the 2 

cointegration space,

[

]

~

5. S t = z t −1 − ~ a − bt

∑ [yˆ

t 2 − a − bt , both the constant and the trend are unrestricted,

T

where z k =

t =1

* M +1,t −1

T

y (m )t −1

]

a and for a k -dimensional random walk, M − r − 1 and vectors ~

[

~ b are determined by regressing the M − r − 1 elements of vector z t = yM* +1,t

]

y (m )t against the

constant and the time trend, and a and b are determined by regressing the squared trend against the trend and the constant. The critical values of the modified TRACE statistic for different numbers of observations (T=50, 80, 100, 200, 300, 400; at 0.05 level of significance) are given in Appendix 3. The simulation results lead to the conclusion that when the percentage of censored observations is high, new critical values are significantly different from the standard ones (see Osterwald-Lenum, 1992). The difference between them is relatively small if 20% or less of observations are censored.

33

Appendix 3. Critical values

Table A3.1. Critical values of the modified trace test for 50 observations. M-r 0.1

0.2

0.3

1 2 3 4 5 6 7 8

4.84 13.98 26.05 41.93 59.97 82.85 104.38 132.19

4.99 13.94 25.98 41.82 59.75 82.56 103.21 130.59

5.06 13.89 25.82 41.51 59.61 82.37 102.37 128.45

1 2 3 4 5 6 7 8

4.76 14.19 25.62 41.85 60.43 82.44 104.34 131.98

4.65 14.06 25.27 40.44 58.56 78.92 104.36 132.23

5.32 13.07 25.08 40.25 57.95 78.52 104.19 127.49

1 2 3 4 5 6 7 8

5.54 13.55 24.60 39.40 56.45 77.02 101.73 126.48

4.40 13.24 23.43 38.18 56.72 77.14 99.75 125.70

4.07 11.80 23.17 38.28 54.67 76.05 99.37 127.73

1 2 3 4 5 6 7 8

7.93 18.33 32.72 49.85 69.22 89.96 116.31 142.95

5.84 18.54 31.00 48.47 66.67 90.71 113.70 143.10

5.40 17.20 29.81 47.40 66.09 88.63 114.10 142.95

1 2 3 4 5 6 7 8

9.44 19.89 34.09 50.00 69.40 90.21 116.61 141.84

8.53 19.01 32.57 49.10 67.85 89.95 113.31 141.28

7.71 18.44 31.99 47.66 66.61 90.07 113.68 142.32

Ratio of censored observations 0.4 0.5 Case 1 4.87 4.72 13.75 13.11 25.13 24.79 41.31 41.11 59.43 59.23 82.25 81.95 101.12 100.03 127.12 126.11 Case 2 5.18 5.29 12.94 12.58 25.13 24.04 38.98 40.02 57.90 56.51 77.51 75.80 102.72 99.98 127.59 125.93 Case 3 3.59 3.41 11.54 11.19 22.41 22.29 36.93 36.02 54.26 52.74 74.22 74.30 96.52 95.44 122.14 122.18 Case 4 4.74 4.38 17.20 15.51 29.41 29.17 46.28 46.03 65.74 64.95 87.69 87.85 111.56 111.20 140.63 138.10 Case 5 7.02 6.23 18.42 16.92 30.54 29.98 47.41 45.87 65.58 64.99 87.53 85.79 111.68 109.53 138.16 135.62

34

0.6

0.7

0.8

4.63 12.35 24.31 40.56 59.01 81.65 98.98 125.43

4.49 12.25 23.87 40.21 58.56 80.92 97.56 123.87

4.20 11.72 23.65 39.98 58.32 80.13 96.12 122.13

4.79 12.44 22.71 39.10 55.59 76.57 99.13 125.37

4.45 12.34 22.37 36.39 54.13 74.45 98.35 123.48

4.17 10.88 22.16 36.27 53.29 72.73 97.30 123.04

2.98 10.49 22.15 34.99 53.03 73.30 95.69 120.83

2.09 10.54 20.57 34.86 50.22 71.18 93.61 120.26

1.61 10.23 19.83 34.51 50.62 70.22 92.27 119.55

4.13 14.85 28.69 43.26 63.78 85.28 112.08 136.58

2.84 14.33 27.22 43.81 62.61 84.07 109.03 138.02

2.12 14.44 26.29 43.36 62.72 83.75 109.29 136.32

5.83 16.07 29.15 43.93 63.55 83.31 109.86 134.78

5.46 15.43 28.12 43.05 61.99 82.17 106.32 135.03

4.22 14.96 27.25 42.20 60.13 80.94 104.44 132.08

Table A3.2. Critical values of the modified trace test for 80 observations. M-r 0.1

0.2

0.3

1 2 3 4 5 6 7 8

5.26 14.15 27.35 42.84 61.12 84.58 109.08 140.85

5.37 14.02 27.13 42.74 59.85 84.32 108.22 139.75

5.42 13.89 26.94 42.57 59.73 84.01 107.29 138.56

1 2 3 4 5 6 7 8

5.26 14.15 27.35 42.84 61.12 84.58 109.08 140.85

4.67 13.73 26.36 42.37 62.05 84.59 109.32 137.70

4.84 13.94 25.63 41.49 59.35 82.83 107.08 135.46

1 2 3 4 5 6 7 8

4.97 14.10 25.49 40.19 59.23 80.50 107.05 133.86

4.88 13.83 25.06 39.29 58.03 80.50 105.70 132.86

4.31 12.65 24.77 38.38 57.71 77.73 103.96 129.46

1 2 3 4 5 6 7 8

7.14 19.15 33.30 51.74 72.70 95.39 124.47 152.45

6.68 18.25 33.46 50.08 70.03 94.80 121.92 151.79

5.78 17.73 31.51 49.29 70.14 93.92 122.21 150.09

1 2 3 4 5 6 7 8

9.41 20.51 36.11 52.80 74.47 96.56 125.59 153.86

9.06 20.29 34.93 51.51 72.35 98.14 121.36 152.66

7.93 19.32 33.36 51.91 71.25 94.39 121.63 149.55

Ratio of censored observations 0.4 0.5 Case 1 5.29 5.21 13.75 13.61 26.78 26.55 42.38 42.21 59.23 58.78 83.71 83.12 106.13 104.87 137.67 136.56 Case 2 5.16 5.19 13.41 12.83 24.82 24.53 40.36 41.59 60.24 57.01 81.17 80.42 106.00 106.12 134.34 131.53 Case 3 3.88 3.30 12.09 12.25 23.53 22.39 38.60 36.91 56.35 54.76 77.48 76.49 101.47 101.38 128.74 128.38 Case 4 4.70 4.12 17.27 16.13 31.34 29.64 48.12 47.74 68.96 67.32 92.30 90.05 117.78 117.98 145.90 146.32 Case 5 7.26 6.76 18.70 17.61 31.73 31.79 50.04 49.16 69.29 68.33 91.86 92.13 118.22 117.46 147.02 146.86

35

0.6

0.7

0.8

5.08 13.48 26.39 42.11 58.51 82.32 103.24 135.43

4.98 13.38 26.15 42.03 58.23 81.45 101.98 134.12

4.72 13.13 25.94 41.85 57.97 80.98 100.48 133.11

4.91 12.94 24.58 39.41 57.24 80.45 104.03 132.89

4.48 11.62 23.66 37.54 55.62 79.10 101.56 131.84

4.63 11.79 23.00 37.97 54.74 75.66 102.61 129.31

2.99 11.28 22.29 36.71 53.52 74.84 100.75 129.44

2.58 10.70 21.24 35.82 54.29 75.38 99.68 126.00

1.73 9.91 21.25 33.30 53.69 73.39 98.44 123.95

4.00 15.25 28.92 47.75 65.50 89.63 117.63 148.64

3.04 15.24 28.46 45.78 66.09 89.17 116.24 145.23

1.99 14.45 28.21 44.91 63.69 89.17 115.65 143.86

6.35 16.91 30.18 47.32 66.90 89.36 116.93 146.20

5.25 16.08 29.40 46.69 66.22 88.09 114.48 143.88

4.97 15.14 29.44 44.52 62.82 86.94 11.34 141.13

Table A3.3. Critical values of the modified trace test for 100 observations. M-r 0.1

0.2

0.3

1 2 3 4 5 6 7 8

5.02 14.37 26.90 43.10 63.25 85.94 112.79 141.99

5.13 14.25 26.72 42.58 62.38 85.78 111.45 140.32

5.21 14.09 26.23 42.11 61.89 84.90 110.34 138.97

1 2 3 4 5 6 7 8

4.38 14.54 26.85 43.43 62.40 86.44 112.59 141.30

4.95 14.61 26.91 42.41 62.01 85.46 111.93 142.93

5.73 14.68 25.71 40.79 61.44 83.82 109.96 138.96

1 2 3 4 5 6 7 8

5.01 13.61 25.95 40.93 60.01 81.76 107.83 135.75

4.58 12.85 24.02 40.67 58.31 79.77 106.31 134.48

4.51 12.61 23.74 38.66 57.86 78.99 105.01 132.59

1 2 3 4 5 6 7 8

7.19 19.04 33.93 51.05 75.44 98.13 126.13 158.41

6.68 18.32 32.79 50.24 73.32 94.12 124.91 154.12

6.19 17.06 32.60 50.09 70.92 94.01 124.12 154.50

1 2 3 4 5 6 7 8

10.15 20.61 35.84 53.75 73.80 100.53 128.21 158.41

9.07 20.35 34.31 52.43 73.63 97.08 126.72 157.32

8.48 20.34 33.82 51.47 71.54 95.96 124.39 156.25

Ratio of censored observations 0.4 0.5 Case 1 5.15 5.13 13.95 13.87 25.89 25.71 41.57 40.39 60.98 59.75 84.12 83.22 108.94 107.56 137.65 135.89 Case 2 4.92 5.29 14.05 13.64 25.02 24.22 42.27 39.44 60.36 59.52 83.11 80.51 109.20 107.93 139.05 135.63 Case 3 4.18 3.42 12.42 11.78 23.26 22.90 39.14 37.33 56.86 55.85 79.36 76.39 104.40 102.80 135.13 130.68 Case 4 5.16 4.09 17.14 16.95 32.01 30.52 48.98 47.08 69.14 68.60 94.25 93.74 120.35 119.08 152.97 150.38 Case 5 7.49 6.82 19.13 18.43 34.07 33.07 51.05 48.96 70.93 70.47 95.11 92.31 122.06 120.28 154.25 151.18

36

0.6

0.7

0.8

5.04 13.71 25.56 39.25 59.11 82.01 106.82 134.42

4.89 13.53 25.11 38.35 57.88 80.48 105.43 133.23

4.62 13.41 23.98 37.57 56.32 79.34 104.58 132.11

5.19 12.73 24.14 39.65 58.54 80.26 105.80 135.49

4.87 12.53 23.19 38.71 56.72 79.86 105.55 132.36

4.53 12.11 22.51 37.35 55.46 77.89 102.94 130.59

2.91 11.09 21.58 37.11 55.10 76.30 100.72 129.27

2.66 11.07 21.96 35.80 54.33 75.15 99.87 126.84

1.56 10.39 19.71 36.13 52.97 74.75 99.52 127.14

3.55 15.57 30.15 47.35 67.77 91.61 118.38 150.42

2.89 14.82 29.95 46.50 66.75 90.46 117.61 146.89

2.28 15.07 28.36 46.34 67.34 90.23 115.19 146.30

6.09 17.37 30.91 48.22 67.14 92.89 118.17 146.94

5.68 16.33 30.41 46.88 67.02 90.14 115.82 147.68

5.26 15.73 29.38 45.04 64.15 89.20 115.20 145.08

Table A3.4. Critical values of the modified trace test for 200 observations. M-r 0.1

0.2

0.3

1 2 3 4 5 6 7 8

5.25 15.12 27.87 44.98 64.87 90.55 116.23 148.90

5.34 15.03 27.71 44.85 64.72 90.01 114.56 139.56

5.46 14.95 27.35 44.73 64.49 88.92 113.21 138.17

1 2 3 4 5 6 7 8

5.11 14.96 27.51 44.39 64.29 90.24 115.76 147.27

5.05 14.92 26.36 43.99 63.27 87.84 115.47 146.31

5.16 14.52 26.03 42.63 63.18 85.45 112.56 144.15

1 2 3 4 5 6 7 8

5.07 14.36 26.73 41.14 61.52 85.99 111.83 142.33

4.40 13.04 24.93 40.89 60.02 84.53 111.29 141.10

4.29 13.03 25.51 40.44 60.67 81.70 108.70 139.05

1 2 3 4 5 6 7 8

7.36 20.21 34.28 53.15 75.79 102.09 132.26 164.85

6.56 18.99 33.51 51.46 74.37 101.25 129.73 161.67

5.96 17.60 33.24 50.08 72.93 99.34 127.25 161.69

1 2 3 4 5 6 7 8

9.95 22.42 36.89 56.33 79.62 104.36 133.42 166.67

9.89 20.71 36.99 55.43 77.22 102.85 132.03 165.43

8.98 20.45 34.77 52.83 74.58 102.08 130.06 163.37

Ratio of censored observations 0.4 0.5 Case 1 5.29 5.11 14.87 14.76 27.12 26.89 44.56 44.51 64.21 64.10 87.64 86.12 111.45 110.35 137.06 135.67 Case 2 5.14 5.80 13.80 14.89 25.61 25.38 44.30 42.04 62.90 60.75 85.03 85.65 112.72 111.86 144.31 142.55 Case 3 4.11 4.07 12.51 12.05 23.58 23.01 39.37 38.62 59.61 57.69 79.20 79.54 108.30 106.59 137.00 136.08 Case 4 4.96 4.38 17.77 16.63 32.29 31.66 51.69 48.94 73.13 70.97 96.87 95.80 127.22 126.42 159.73 158.92 Case 5 7.81 7.84 19.06 18.81 34.27 33.53 51.65 51.63 74.26 72.19 98.67 98.41 128.35 127.20 161.79 158.70

37

0.6

0.7

0.8

5.03 13.99 26.65 43.98 63.78 85.78 109.12 134.89

4.87 13.76 26.45 43.73 63.65 84.53 108.08 133.22

4.75 13.12 26.31 42.21 63.49 82.90 107.12 132.89

5.35 13.48 25.27 40.84 58.98 82.47 108.55 141.23

5.16 13.29 24.50 40.65 58.15 80.51 107.13 138.90

5.08 12.50 23.18 38.10 57.59 80.95 107.04 136.74

3.12 11.49 22.09 37.30 56.10 77.60 104.83 135.30

2.56 11.15 21.59 36.59 55.04 77.03 102.88 133.73

1.77 10.39 21.35 35.81 53.88 75.68 103.54 131.19

4.15 16.13 30.58 47.14 69.36 94.43 125.86 157.76

3.46 15.65 29.82 47.26 68.29 92.06 122.60 155.15

2.39 14.48 29.15 46.39 67.80 92.25 120.40 151.23

6.86 18.63 32.27 51.99 72.28 97.85 125.18 157.31

6.48 17.97 30.98 48.44 70.85 94.85 124.43 152.20

5.83 16.74 30.37 47.87 69.33 91.83 120.24 151.81

Table A3.5. Critical values of the modified trace test for 300 observations. M-r 0.1

0.2

0.3

1 2 3 4 5 6 7 8

5.15 14.57 28.39 44.89 65.93 88.12 117.01 148.23

5.23 14.43 28.01 44.56 65.13 87.22 116.02 146.57

5.31 14.01 27.66 44.31 64.22 86.45 114.58 145.12

1 2 3 4 5 6 7 8

5.11 14.47 28.24 44.68 64.90 88.09 116.88 149.12

4.96 14.80 28.09 43.06 64.47 89.85 116.92 148.92

5.43 14.52 27.14 45.44 63.77 87.73 115.25 146.27

1 2 3 4 5 6 7 8

4.91 13.87 26.43 41.14 62.03 85.65 112.32 145.46

4.54 13.53 25.71 41.27 60.82 84.52 110.69 142.94

4.59 12.65 25.00 39.96 60.53 82.71 110.33 139.33

1 2 3 4 5 6 7 8

8.00 20.29 35.17 53.49 76.19 103.24 133.40 167.27

6.77 18.85 33.90 52.15 75.78 102.49 131.64 164.88

5.85 19.19 34.46 51.44 74.10 100.61 129.77 163.46

1 2 3 4 5 6 7 8

9.98 22.27 38.41 55.51 78.58 104.55 135.25 169.96

9.50 21.92 36.74 56.03 76.71 103.88 133.90 169.59

8.85 20.07 36.33 52.47 76.46 100.89 132.61 167.90

Ratio of censored observations 0.4 0.5 Case 1 5.26 5.19 13.56 13.31 27.12 26.74 44.01 43.24 63.11 62.25 85.11 83.45 113.24 112.16 143.98 142.54 Case 2 5.23 5.67 13.96 14.61 26.60 25.72 41.81 41.57 62.38 62.42 87.14 84.69 114.34 113.43 146.42 143.56 Case 3 4.67 3.62 12.18 11.55 25.25 23.24 39.73 38.45 58.98 58.18 81.97 81.51 107.73 109.36 138.34 137.95 Case 4 5.41 4.67 17.81 16.84 33.94 32.41 50.18 49.61 73.25 70.14 99.79 97.51 128.36 126.92 162.53 159.87 Case 5 8.43 7.46 18.98 19.20 35.09 33.60 52.90 52.09 75.73 74.17 100.57 101.12 132.03 128.69 165.24 161.25

38

0.6

0.7

0.8

5.03 13.19 26.15 41.98 61.09 82.10 110.89 141.29

4.13 12.87 26.01 41.23 59.87 80.76 109.72 140.17

3.78 12.65 25.43 40.56 59.13 80.14 108.56 138.87

5.45 13.76 25.53 41.27 60.37 84.36 110.68 141.24

5.63 13.27 24.87 40.50 61.01 82.22 109.51 140.58

5.19 12.76 24.89 39.27 59.63 81.78 107.59 140.24

3.21 10.94 22.04 38.02 56.64 79.53 107.38 138.17

2.68 10.91 21.86 36.17 55.53 77.94 103.31 134.80

1.77 10.63 21.56 36.24 55.31 77.37 102.50 133.75

4.13 16.17 29.97 48.48 71.72 96.72 126.42 159.67

2.79 15.84 30.83 48.23 69.60 93.27 122.66 158.87

2.42 14.62 28.45 46.87 67.94 94.49 121.47 154.01

7.33 18.92 32.27 51.05 73.40 98.45 127.98 161.18

7.30 17.00 31.72 49.52 71.10 94.91 125.19 159.24

6.53 17.12 30.92 48.71 69.63 94.29 124.00 154.44

Table A3.6. Critical values of the modified trace test for 400 observations. M-r 0.1

0.2

0.3

1 2 3 4 5 6 7 8

5.11 14.55 28.35 44.85 65.95 88.15 117.05 148.22

5.24 14.46 28.02 44.52 65.11 87.14 116.01 146.51

5.31 14.03 27.64 44.36 64.25 86.32 114.54 145.16

1 2 3 4 5 6 7 8

4.85 14.78 27.99 44.81 66.36 89.82 119.11 114.91

4.87 15.09 27.28 43.63 64.77 88.79 118.50 113.97

5.46 14.19 26.66 43.88 63.79 89.96 116.60 112.18

1 2 3 4 5 6 7 8

5.06 13.70 26.68 42.03 61.64 84.51 113.57 145.53

4.85 13.46 25.52 41.28 61.30 83.58 112.02 143.28

4.20 12.33 25.07 40.77 59.64 82.42 110.23 142.38

1 2 3 4 5 6 7 8

7.80 19.76 34.62 55.22 78.81 104.75 133.40 169.25

6.99 19.13 34.35 53.05 76.02 101.77 133.33 165.34

6.02 18.36 32.65 52.84 75.14 101.26 130.16 165.57

1 2 3 4 5 6 7 8

9.91 22.42 38.07 58.27 80.91 106.15 137.27 172.57

10.19 21.48 36.76 56.49 78.69 105.78 136.67 170.04

8.52 20.86 35.64 53.48 77.76 103.83 135.20 166.77

Ratio of censored observations 0.4 0.5 Case 1 5.25 5.20 13.58 13.37 27.11 26.75 44.05 43.23 63.13 62.21 85.09 83.32 113.19 112.16 143.94 142.52 Case 2 5.58 5.54 14.67 13.55 26.62 26.23 42.11 41.66 62.58 62.90 86.67 86.35 114.91 113.97 109.55 108.31 Case 3 4.25 3.76 12.75 11.34 23.59 23.91 40.16 39.65 60.39 57.17 83.38 82.18 110.32 107.06 140.25 139.30 Case 4 4.83 5.21 18.59 17.29 31.95 30.92 51.91 50.91 74.35 73.67 99.32 99.38 128.15 124.35 163.83 160.58 Case 5 8.86 8.01 19.80 19.35 35.22 34.52 52.07 52.41 76.92 73.53 101.99 101.11 130.89 128.77 165.43 163.37

39

0.6

0.7

0.8

5.07 13.23 26.12 41.91 61.03 82.23 110.92 141.23

4.25 12.86 26.03 41.22 59.81 80.65 109.75 140.11

3.98 12.66 25.41 40.51 59.12 80.12 108.51 138.86

5.89 13.42 25.22 41.00 60.83 85.15 112.18 107.55

6.16 13.62 25.00 39.38 60.78 84.12 109.55 105.13

5.59 13.50 24.93 39.76 58.21 82.93 108.31 104.62

3.68 11.63 23.11 37.73 56.54 81.83 105.35 137.75

2.77 10.94 22.39 37.02 55.33 77.86 103.80 136.01

1.72 10.68 21.37 35.80 54.10 77.91 104.46 134.13

3.97 16.12 31.16 48.71 70.04 95.24 126.99 161.26

3.11 15.39 30.84 48.50 70.33 97.11 125.48 159.60

2.05 14.80 29.45 47.32 68.56 95.13 124.10 156.34

7.33 18.25 33.28 50.78 73.61 101.10 127.50 161.66

6.97 18.15 32.73 50.28 73.33 96.63 125.56 158.92

6.30 17.75 31.13 49.00 69.75 93.68 123.86 156.66

Tables

Table 1. Results of testing order of integration of variables on the basis of Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) test

Variable ext idt-idtf Dt-Dtf BCt NFAt pt-ptf It-Itf xt-xtf m2t-m2tf tott

Level ADF p-value PP p-value 0.16 0.18 0.27 0.41 0.79 0.77 0.17 0.11 0.35 0.40 0.56 0.61 0.63 0.40 0.46 0.57 0.62 0.47 0.00 0.00

First difference ADF p-value PP p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -

Decision I(1) I(1) I(1) I(1) I(1) I(1) I(1) I(1) I(1) I(0)

Table 2. Cointegration rank test

Number of cointegrating vectors 0 1 2 3

TRACE statistic

Simulated critical values

85.92 61.25 41.33 18.75

80.37 59.16 40.07 25.43

40

Table 3. Diagnostic tests for residuals (p-values in the brackets)

Equation

ARCH(6)

Normality S-W

∆Nt* ∆ext

5.67(0.46) 6.45(0.37)

0.96 0.95

∆(pt-ptf)

4.87(0.56)

0.94

∆(xt-xtf)

2.17(0.90)

0.97

∆(idt-idtf)

5.98(0.43)

0.96

∆(m2t-m2tf)

5.56(0.47)

0.96

Autocorrelation LM(12)

10.11(0.61)

Table 4. Measuring goodness of fit and ex-post forecast errors

Equation explaining instability Tobit-CVAR Standard Tobit model ME1 0.91 0.84 ME2 0.82 0.73 Ex-post forecast errors when predicting exchange rate Full model Model without currency market instability RMSE 0.031 0.066 MAPE 0.019 0.034

41

- A pioneering approach to exchange rate modelling is proposed. - A novel measure of currency market disequilibrium is established. - The impact of market disequilibrium on the exchange rate is empirically confirmed. - A concept of the Tobit cointegrated vector autoregressive model is introduced. - A method of inference in the system with nonstationary censored variables is developed.