**1. Introduction**

The risk exposures from the US during the 2007–2009 financial crisis spread rapidly to the global financial markets and gradually increased its severity while the great number of banks bankrupted due to increase in interest rates. The major role for the bankruptcies of the banks in European countries was the domino effect of the spread of interest rate risk in the interbank market and liquidity risk (e.g., [1, 2]). Since the liquidity risks correspond to counterparty risk, the idiosyncratic credit problems arising from the US subprime mortgage market spread rapidly to other countries through the channels of changes in interest rate (e.g., [2–5]). Brunnermeier and Pedersen [3] empirically show that increase in interest rates affect the financial institutions that have liquidity problems as those institutions are more open to risk contagion arising from the interest rate rise. "For this reason, banks, which carry interbank credit risk threats, are exposed to liquidity risks and such a systemic risk contagion causes subsequent bankruptcies (see e.g., [6–11])" ([12], p. 243).

Even though the existing literature mainly addresses the issues such as risk contagion across stock markets or foreign exchange markets due to counterparty relationships, macroeconomic risk or financial linkages; how interest rate risk propagates around global financial markets is not fully investigated (e.g., [10, 13–15]). As interest rates can be used domestically to absorb the external shocks and to balance the currency, the propagation of the interest rate risks between financial markets gains much more importance for economically semi- and fully open countries.

How foreign interest rates and the exchange rates affect the domestic interest rates can be shown with the following equation:

$$\dot{\mathbf{u}}\_t = \dot{\mathbf{i}}\_t^\* + E\_t \mathbf{e}\_{t+1} - \mathbf{e}\_t + \pi\_t \tag{1}$$

This chapter is motivated to some extent by the earlier work of Edwards [26] and Borensztein et al. [21]. Therefore, following those studies, in this study, I aim to construct a vector auto regression (VAR) model to examine the effect of external shocks on Turkish short-term interest rates and the exchange rate. Differently from Edwards [26] and Borensztein et al. [21] and the more recent papers, I investigate how the impact of external shocks change according to tranquil and politically stressed periods as Turkey is a rather politically instable country and this situation causes authorities to interfere with the floating exchange rate regime every now

*External Factors on Turkish Short-Term Interest Rates and Daily Exchange Rates: Tranquil…*

The chapter is structured as follows. Section 2 presents data we use to analyze the impacts of external shocks on the Turkish short-term interest rates and the exchange rate; and the VAR model under Section 2.2. Section 3 reports the estimation results according to full period, each politically stressed periods and the politically tranquil periods. Finally, the wrap up of the results and conclusions are

The time period, in this study, is determined as January 2011–December 2018. Turkey has been governed by one political party since 2002. Therefore, the period from 2002 to today can be counted as a rather politically stable period for Turkey. However, in our study we do not want to include the first 5 years of the AKP (The Justice and Development Party) governments as this period can be counted as the rebalancing and redevelopment period after the heavy financial crisis of 2001. Furthermore, as during the years between 2007 and 2010, global financial crisis may have a dominant role on the markets instead of local developments, we do not include this period into our study too. Therefore, the period that we decide to examine, 2011–2018, solely shows us the impact of external factors change on the short-term interest rates and the daily exchange rates according to politically

In this study, 3-months interbank rates are used as the short-term interest rates.

Data set covers the period January 1, 2011–December 31, 2018 for a total of 2087 daily observations and is downloaded from Bloomberg Terminal. **Figure 1** repre-

In **Figure 1**, the first graph presents how USD/TRY exchange rate changes between 2011 and 2019. Second and third graphs present the pattern of short-term interest rates for USA and Turkey between 2011 and 2019. The final graph shows how emerging market risk premium changes during the 2011–2019 period. To be able to determine the politically stressed times that have significant impacts on the financial markets of Turkey, we identify the financial stress periods. For this purpose, we first identify the anomalies on the daily price changes on Borsa

To be able to assess how group of emerging countries affect Turkish domestic interest rates and the exchange rates, daily iShares MSCI emerging markets ETF is used as the proxy of the emerging market risk premia (difference between return of

a risky asset and the risk-free rate). Finally for the exchange rate, daily spot exchange rate against the US Dollar is used. Therefore, our data set includes daily 3-months interbank interest rates for Turkey and the US, daily iShares MSCI emerging markets ETF and daily spot exchange rate against the US Dollar for

sents the graphs of each group of data for the examined period.

and then.

**2.1 Data**

Turkish Lira.

**81**

offered in Section 4.

**2. Data and methodology**

*DOI: http://dx.doi.org/10.5772/intechopen.89931*

stressed times or tranquil periods.

where *it* is the domestic interest rate with maturity *t +* 1*; i* <sup>∗</sup> *<sup>t</sup>* is the foreign interest rates with the same maturity; *et* is the natural logarithm of the spot exchange rate at time *t*; and *π<sup>t</sup>* is the country risk premium. According to this, every shock to *i* <sup>∗</sup> or *π* can be absorbed by changes in domestic interest rates and changes in the expected rate of depreciation. Therefore, it is possible to say that under floating exchange rate regime, policy makers have freedom to increase or decrease the domestic interest rates to adjust the exchange rate. For example, "a positive shock to *i* <sup>∗</sup> or *π* may cause an immediate devaluation of the exchange rate which overshoots its long-run equilibrium and tends to appreciate (or reduce its rate of depreciation)" ([16], p. 7). In other words, under floating exchange rate regime, flexible exchange rate can absorb external shocks.

According to Edwards [17], the interest rate spread, which can be defined as the difference between lending and riskless rates, is a key transmission channel for interest rate risk propagation (e.g., [5, 18–20]). Borensztein et al. [21] examine the impact of international interest rate shocks and emerging market risk premia on domestic interest rates and exchange rates for both emerging and developed countries. The authors find different results for Latin American and Asian economies and for different exchange rate regimes. According to that in Mexico and Argentina, emerging market risk premia significantly affects the interest rates. On the other hand, the Asian countries show different reactions according to their exchange rate regimes; Singapore which has a floating exchange rate regime seems unaffected by the external shocks while Hong Kong responses significantly to the emerging market risk premia.

There are more recent papers that investigate the impacts of external shocks for various countries. In Ref. [22], Demirel investigates the impulse responses of the Turkish economy to the US interest rate shocks. The study reveals that Turkey is less sensitive to the interest rate shocks while she has lower levels of external debt. Therefore, the author concludes that the foreign interest rate shocks depend on the level of external debt for small-open economies. Allegret et al. [23] examine the relative importance of external shocks in domestic fluctuations for East Asian countries. Using a structural VAR model, the authors show that real oil price and the US GDP shocks have significant impacts on domestic activity. They also reveal that since the mid-1990s, external shocks have rising impacts on domestic variables in those countries. Using a trend-cycle VAR model, Andrle et al. [24] investigate how external factors affect the Poland's domestic variables. According to that, the authors reach the conclusion that about 50% of Poland's output and interest rate variance and about 25% of the variance of inflation can be explained with shocks from Euro zone. Pelipas et al. [25] test the significance of Russia's GDP and oil prices as the external factors on Belarus' economy. Using generalized impulse response functions, the authors show that oil prices have strong and negative impact on the economy while Russia's GDP does not have that strong impact.

*External Factors on Turkish Short-Term Interest Rates and Daily Exchange Rates: Tranquil… DOI: http://dx.doi.org/10.5772/intechopen.89931*

This chapter is motivated to some extent by the earlier work of Edwards [26] and Borensztein et al. [21]. Therefore, following those studies, in this study, I aim to construct a vector auto regression (VAR) model to examine the effect of external shocks on Turkish short-term interest rates and the exchange rate. Differently from Edwards [26] and Borensztein et al. [21] and the more recent papers, I investigate how the impact of external shocks change according to tranquil and politically stressed periods as Turkey is a rather politically instable country and this situation causes authorities to interfere with the floating exchange rate regime every now and then.

The chapter is structured as follows. Section 2 presents data we use to analyze the impacts of external shocks on the Turkish short-term interest rates and the exchange rate; and the VAR model under Section 2.2. Section 3 reports the estimation results according to full period, each politically stressed periods and the politically tranquil periods. Finally, the wrap up of the results and conclusions are offered in Section 4.

### **2. Data and methodology**

#### **2.1 Data**

Even though the existing literature mainly addresses the issues such as risk contagion across stock markets or foreign exchange markets due to counterparty relationships, macroeconomic risk or financial linkages; how interest rate risk propagates around global financial markets is not fully investigated (e.g., [10, 13–15]). As interest rates can be used domestically to absorb the external shocks and to balance the currency, the propagation of the interest rate risks between financial markets gains much more importance for economically semi- and fully open countries. How foreign interest rates and the exchange rates affect the domestic interest

rates with the same maturity; *et* is the natural logarithm of the spot exchange rate at time *t*; and *π<sup>t</sup>* is the country risk premium. According to this, every shock to *i* <sup>∗</sup> or *π* can be absorbed by changes in domestic interest rates and changes in the expected rate of depreciation. Therefore, it is possible to say that under floating exchange rate regime, policy makers have freedom to increase or decrease the domestic interest

cause an immediate devaluation of the exchange rate which overshoots its long-run equilibrium and tends to appreciate (or reduce its rate of depreciation)" ([16], p. 7). In other words, under floating exchange rate regime, flexible exchange rate can

According to Edwards [17], the interest rate spread, which can be defined as the

difference between lending and riskless rates, is a key transmission channel for interest rate risk propagation (e.g., [5, 18–20]). Borensztein et al. [21] examine the impact of international interest rate shocks and emerging market risk premia on domestic interest rates and exchange rates for both emerging and developed countries. The authors find different results for Latin American and Asian economies and for different exchange rate regimes. According to that in Mexico and Argentina, emerging market risk premia significantly affects the interest rates. On the other hand, the Asian countries show different reactions according to their

exchange rate regimes; Singapore which has a floating exchange rate regime seems unaffected by the external shocks while Hong Kong responses significantly to the

economy while Russia's GDP does not have that strong impact.

There are more recent papers that investigate the impacts of external shocks for various countries. In Ref. [22], Demirel investigates the impulse responses of the Turkish economy to the US interest rate shocks. The study reveals that Turkey is less sensitive to the interest rate shocks while she has lower levels of external debt. Therefore, the author concludes that the foreign interest rate shocks depend on the level of external debt for small-open economies. Allegret et al. [23] examine the relative importance of external shocks in domestic fluctuations for East Asian countries. Using a structural VAR model, the authors show that real oil price and the US GDP shocks have significant impacts on domestic activity. They also reveal that since the mid-1990s, external shocks have rising impacts on domestic variables in those countries. Using a trend-cycle VAR model, Andrle et al. [24] investigate how external factors affect the Poland's domestic variables. According to that, the authors reach the conclusion that about 50% of Poland's output and interest rate variance and about 25% of the variance of inflation can be explained with shocks from Euro zone. Pelipas et al. [25] test the significance of Russia's GDP and oil prices as the external factors on Belarus' economy. Using generalized impulse response functions, the authors show that oil prices have strong and negative impact on the

*<sup>t</sup>* þ *Etet*þ<sup>1</sup> � *et* þ *π<sup>t</sup>* (1)

*<sup>t</sup>* is the foreign interest

<sup>∗</sup> or *π* may

rates can be shown with the following equation:

*Financial Crises - A Selection of Readings*

absorb external shocks.

emerging market risk premia.

**80**

*it* ¼ *i* ∗

where *it* is the domestic interest rate with maturity *t +* 1*; i* <sup>∗</sup>

rates to adjust the exchange rate. For example, "a positive shock to *i*

The time period, in this study, is determined as January 2011–December 2018. Turkey has been governed by one political party since 2002. Therefore, the period from 2002 to today can be counted as a rather politically stable period for Turkey. However, in our study we do not want to include the first 5 years of the AKP (The Justice and Development Party) governments as this period can be counted as the rebalancing and redevelopment period after the heavy financial crisis of 2001. Furthermore, as during the years between 2007 and 2010, global financial crisis may have a dominant role on the markets instead of local developments, we do not include this period into our study too. Therefore, the period that we decide to examine, 2011–2018, solely shows us the impact of external factors change on the short-term interest rates and the daily exchange rates according to politically stressed times or tranquil periods.

In this study, 3-months interbank rates are used as the short-term interest rates. To be able to assess how group of emerging countries affect Turkish domestic interest rates and the exchange rates, daily iShares MSCI emerging markets ETF is used as the proxy of the emerging market risk premia (difference between return of a risky asset and the risk-free rate). Finally for the exchange rate, daily spot exchange rate against the US Dollar is used. Therefore, our data set includes daily 3-months interbank interest rates for Turkey and the US, daily iShares MSCI emerging markets ETF and daily spot exchange rate against the US Dollar for Turkish Lira.

Data set covers the period January 1, 2011–December 31, 2018 for a total of 2087 daily observations and is downloaded from Bloomberg Terminal. **Figure 1** represents the graphs of each group of data for the examined period.

In **Figure 1**, the first graph presents how USD/TRY exchange rate changes between 2011 and 2019. Second and third graphs present the pattern of short-term interest rates for USA and Turkey between 2011 and 2019. The final graph shows how emerging market risk premium changes during the 2011–2019 period.

To be able to determine the politically stressed times that have significant impacts on the financial markets of Turkey, we identify the financial stress periods. For this purpose, we first identify the anomalies on the daily price changes on Borsa

• the period which starts with the 7th June 2015 general elections and continues

*External Factors on Turkish Short-Term Interest Rates and Daily Exchange Rates: Tranquil…*

• 2 weeks period which starts with the shootdown of Russian plane on Turkish

• 2 weeks period which starts with the military coup attempt on 15th July 2016,

• 3 months period which starts with the announcement of new cabinet on 2nd July 2018 and strengthens with Pastor Branson's house arrest and ends with

To be able to show the relation between short-term interest rates of USA, shortterm interest rates of Turkey and USD/TRY exchange rate, we prepare **Figure 2**. Although the figure does not allow us to statistically prove the correlation between the US interest rates, Turkish interest rates and USD/TRY exchange rate; it is still possible to see that especially in the latter period (after 2017) short-term interest rates of USA, short-term interest rates of Turkey and USD/TRY share significant

**Figure 2** presents the graphs of the short-term interest rates for both US and

interest rates of Turkey and USD/TRY exchange rate together to show whether Turkish risky assets and emerging markets risk premia share common pattern

Differently from **Figure 2**, **Figure 3** brings emerging risk premia and short-term

**Figure 3** presents the graphs of iShares MSCI Emerging Markets ETF, Turkish short-term interest rate and USD/TRY exchange rate for the 2011–2019 period.

In this chapter, to be able to examine the effect of US interest rates and emerging

market risk premia on the domestic short-term interest rate of Turkey and exchange rate against the US Dollar, we construct a vector auto regression (VAR) model. More specifically, the model includes the Turkish short-term interest rate, the US short-term interest rate, the natural logarithm of the exchange rate against the US dollar and iShares MSCI emerging markets ETF. We expect to see that during the tranquil periods, the Turkish short-term interest rate and the USD/TRY exchange rate are both positively and significantly affected by the US short-term interest rate and the emerging market risk premia. According to that, we expect to see that short-term Turkish interest rate and the USD/TRY exchange rate increase with the increasing short-term US interest rate and the emerging market risk

*Short-term interest rate of USA, short-term interest rate of Turkey and USD/TRY exchange rate.*

Turkey and the USD/TRY exchange rate for the 2011–2019 period.

until the announcement of new elections on 25th August 2015,

border on 24th November 2015,

*DOI: http://dx.doi.org/10.5772/intechopen.89931*

Brunson's return to USA.

common pattern.

**2.2 Methodology**

**Figure 2.**

**83**

during the examined period.

**Figure 1.**

*Exchange rates in Turkey, interest rates in Turkey and the US and emerging markets risk premia (2011–2019).*

Istanbul. We define the anomalies as 5% or above drop on the main index of Borsa Istanbul in total in at least 5 days period.

Financial markets experience either a crash or a bear market. The widely used criteria for a crash is 10% drop from the peak prices in 1 or 2 days and a drop of at least 20% off peak prices in a wider time period for the bear market. While identifying the price anomalies in Turkish market, we consider both of those criteria. To be able to decide on the exact drop rate, we examine the sharp drops in Borsa Istanbul for 20 years period. During that period, the average correction rate for the market is calculated as 3.7%. Therefore to able to identify a price movement as an anomaly, we need to determine a rate above this rate. However, as we do not want to keep that rate as high as a rate that is needed to classify the drop as a stock crash, we identify 5% and above rates as price anomalies. Finally, as we determine that Borsa Istanbul shows the strongest reactions to negative events or news in the first 5 days on average, we decide on the 5 days criterion.

After the examination of the daily price changes of Borsa Istanbul from 2011 till 2019, we identify eight different periods that BIST100 lose at least 5% in total in at least 5 days.

Following the identification of the financial stress periods, we identify the domestic political developments that occur on the same periods. These are;


*External Factors on Turkish Short-Term Interest Rates and Daily Exchange Rates: Tranquil… DOI: http://dx.doi.org/10.5772/intechopen.89931*


To be able to show the relation between short-term interest rates of USA, shortterm interest rates of Turkey and USD/TRY exchange rate, we prepare **Figure 2**. Although the figure does not allow us to statistically prove the correlation between the US interest rates, Turkish interest rates and USD/TRY exchange rate; it is still possible to see that especially in the latter period (after 2017) short-term interest rates of USA, short-term interest rates of Turkey and USD/TRY share significant common pattern.

**Figure 2** presents the graphs of the short-term interest rates for both US and Turkey and the USD/TRY exchange rate for the 2011–2019 period.

Differently from **Figure 2**, **Figure 3** brings emerging risk premia and short-term interest rates of Turkey and USD/TRY exchange rate together to show whether Turkish risky assets and emerging markets risk premia share common pattern during the examined period.

**Figure 3** presents the graphs of iShares MSCI Emerging Markets ETF, Turkish short-term interest rate and USD/TRY exchange rate for the 2011–2019 period.

#### **2.2 Methodology**

Istanbul. We define the anomalies as 5% or above drop on the main index of Borsa

*Exchange rates in Turkey, interest rates in Turkey and the US and emerging markets risk premia (2011–2019).*

Financial markets experience either a crash or a bear market. The widely used criteria for a crash is 10% drop from the peak prices in 1 or 2 days and a drop of at least 20% off peak prices in a wider time period for the bear market. While identifying the price anomalies in Turkish market, we consider both of those criteria. To be able to decide on the exact drop rate, we examine the sharp drops in Borsa Istanbul for 20 years period. During that period, the average correction rate for the market is calculated as 3.7%. Therefore to able to identify a price movement as an anomaly, we need to determine a rate above this rate. However, as we do not want to keep that rate as high as a rate that is needed to classify the drop as a stock crash, we identify 5% and above rates as price anomalies. Finally, as we determine that Borsa Istanbul shows the strongest reactions to negative events or news in the first

After the examination of the daily price changes of Borsa Istanbul from 2011 till 2019, we identify eight different periods that BIST100 lose at least 5% in total in at

Following the identification of the financial stress periods, we identify the domestic political developments that occur on the same periods. These are;

• 3 weeks period which starts with Gezi Park incidents on 28th May 2013,

• 10 days period which starts with the early retirement request of commanders

• 1 month period which starts with the operation of FETO terror organization to

• 10 days period which starts on 31st July 2014 prior to presidential election,

Istanbul in total in at least 5 days period.

*Financial Crises - A Selection of Readings*

5 days on average, we decide on the 5 days criterion.

of Turkish Army on 29th July 2011,

government authorities on 17th December 2013,

least 5 days.

**82**

**Figure 1.**

In this chapter, to be able to examine the effect of US interest rates and emerging market risk premia on the domestic short-term interest rate of Turkey and exchange rate against the US Dollar, we construct a vector auto regression (VAR) model. More specifically, the model includes the Turkish short-term interest rate, the US short-term interest rate, the natural logarithm of the exchange rate against the US dollar and iShares MSCI emerging markets ETF. We expect to see that during the tranquil periods, the Turkish short-term interest rate and the USD/TRY exchange rate are both positively and significantly affected by the US short-term interest rate and the emerging market risk premia. According to that, we expect to see that short-term Turkish interest rate and the USD/TRY exchange rate increase with the increasing short-term US interest rate and the emerging market risk

**Figure 2.**

*Short-term interest rate of USA, short-term interest rate of Turkey and USD/TRY exchange rate.*

*det In* � *<sup>Π</sup>*1*<sup>z</sup>* � … � *<sup>Π</sup>pzp* <sup>¼</sup> <sup>0</sup> (6)

*c* (7)

lie outside the complex unit circle (have modulus greater than one), or, equiva-

lently, if the eigenvalues of the companion matrix have modules less than one. Assuming that the process has been initialized in the infinite past, then a stable VAR (p) process is stationary and ergodic with time invariant means, variances and

*External Factors on Turkish Short-Term Interest Rates and Daily Exchange Rates: Tranquil…*

If *Yt* is covariance stationary, then the unconditional mean is given by;

μ ¼ *In* � *Π*<sup>1</sup> � … � *Π<sup>p</sup>* �<sup>1</sup>

*Yt* � <sup>μ</sup> <sup>¼</sup> *<sup>Π</sup>*1ð Þþ *Yt*�<sup>1</sup> � <sup>μ</sup> *<sup>Π</sup>*2ð Þþ *Yt*�<sup>2</sup> � <sup>μ</sup> … <sup>þ</sup> *<sup>Π</sup><sup>p</sup> Yt*�*<sup>p</sup>* � <sup>μ</sup> <sup>þ</sup> *<sup>ε</sup><sup>t</sup>* (8)

The general form of the VAR(p) model with deterministic terms and exogenous

where *Dt* represents an *(l* � *1)* matrix of deterministic components, *Xt* repre-

To be able to estimate the basic VAR(p) model, each equation in the model can

where *yi* is a (*T* � *1*) vector of observations on the *i*th equation, *Z* is a *(T* � *k)*

VAR(p) is in the form of a SUR model where each equation has the same explanatory variables, each equation may be estimated separately by ordinary least squares

In this study, to be able to show how the impact of external shocks on domestic interest rates and the exchange rates change according to political stress in Turkey, we construct a VAR model. **Figure 4** plots interest rates and exchange rates in Turkey over the period of 2011–2019. The politically stressed periods are

highlighted on this figure to show how domestic interest rates and exchange rates react to political developments. Therefore, eight shaded lines on **Figure 4** identify

• 10 days period which starts with the early retirement request of commanders

• 1 month period which starts with the operation of FETO terror organization to

• 3 weeks period which starts with Gezi Park incidents on 28th May 2013,

*<sup>t</sup>*�1, … , *<sup>Y</sup>*<sup>0</sup>

*t*�*p*

sents an *(m* � *1)* matrix of exogenous variables, and ɸ and *G* are parameter

*<sup>t</sup>* ¼ 1, *Y*<sup>0</sup>

vector of parameters and *ei* is a *(T* � *1)* error with covariance matrix *<sup>σ</sup>*<sup>2</sup>

without losing efficiency relative to generalized least squares.

the government authorities on 17th December 2013,

the following major political crises in Turkey;

of Turkish Army on 29th July 2011,

*Yt* ¼ *Π*1*Yt*�<sup>1</sup> þ *Π*2*Yt*�<sup>2</sup> þ … þ *ΠpYt*�*<sup>p</sup>* þ ɸ*Dt* þ *GXt* þ *ε<sup>t</sup>* (9)

*yi* ¼ *Zπ<sup>i</sup>* þ *ei*, *i* ¼ 1, … , *n* (10)

, *<sup>k</sup>* <sup>¼</sup> *np* <sup>þ</sup> 1, *<sup>π</sup><sup>i</sup>* is a (k � 1)

*<sup>i</sup> IT*. Since the

The mean-adjusted form of the VAR(p) is then;

*DOI: http://dx.doi.org/10.5772/intechopen.89931*

autocovariances.

variables is given by;

matrices.

be written as;

**3. Results**

**85**

matrix with *t*th row given by *Z*<sup>0</sup>

**Figure 3.** *Emerging markets risk premia, short-term interest rate of Turkey and USD/TRY.*

premia. However, during the politically stressed periods it is not possible to estimate the relations between those variables as each political stress may have a different impact according to their dynamics. For instance, while a fully domestic political stress may cause Turkish financial markets to separate from the rest of the world, a political stress that is caused by an international development may cause Turkish markets to more sensitive to the external shocks.

"The vector autoregression (VAR) model is one of the most successful, flexible, and easy to use models for the analysis of multivariate time series" ([27], p. 385). It is a natural extension of the univariate autoregressive model to dynamic multivariate time series.<sup>1</sup>

Let *Yt = (y1t, y2t, … , ynt)'* denote an *(n* � *1)* vector of time series variables. The basic *p-lag* vector autoregressive model has the form;

$$Y\_t = \mathbf{c} + \Pi\_1 Y\_{t-1} + \Pi\_2 Y\_{t-2} + \dots + + \Pi\_p Y\_{t-p} + e\_t, \ t = 1, \dots, T \tag{2}$$

where *Π<sup>i</sup>* are *(n* � *n)* coefficient matrices and *ε<sup>t</sup>* is an *(n* � *1)* unobservable zero mean White noise vector process with time invariant covariance matrix Σ. According to this, for example, a bivariate VAR(2) model equation by equation has the form;

$$\mathbf{y}\_{1t} = \mathbf{c}\_{1} + \boldsymbol{\pi}\_{11}^{1}\mathbf{Y}\_{1t-1} + \boldsymbol{\pi}\_{12}^{1}\mathbf{Y}\_{2t-1} + \boldsymbol{\pi}\_{11}^{1}\mathbf{Y}\_{1t-2} + \boldsymbol{\pi}\_{12}^{2}\mathbf{Y}\_{2t-2} + \mathbf{e}\_{1t} \tag{3}$$

$$y\_{2t} = \varepsilon\_1 + \pi\_{21}^1 Y\_{1t-1} + \pi\_{22}^1 Y\_{2t-1} + \pi\_{21}^1 Y\_{1t-1} + \pi\_{22}^2 Y\_{2t-1} + \varepsilon\_{2t} \tag{4}$$

where *cov(ε*1*<sup>t</sup>*, *ε*2*<sup>t</sup>*Þ ¼ *σ*12. Each equation has the same regressors-lagged values of *y*1*<sup>t</sup>* and *y*2*<sup>t</sup>* . Hence the VAR(p) model is just a seemingly unrelated regression (SUR) model with lagged variables and deterministic terms as common regressors.

In lag operator notation, the VAR(p) is written as;

$$H(L)Y\_t = \mathbf{c} + \mathbf{e}\_t \tag{5}$$

where *<sup>Π</sup>*ð Þ¼ *<sup>L</sup> In* � *<sup>Π</sup>*1*<sup>L</sup>* � … � *<sup>Π</sup>pL<sup>p</sup>*. The VAR(p) is stable if the roots of

<sup>1</sup> The theoretical presentation of vector autoregressive models that is used in the Methodology part is taken from the book of Zivot and Wang [26].

*External Factors on Turkish Short-Term Interest Rates and Daily Exchange Rates: Tranquil… DOI: http://dx.doi.org/10.5772/intechopen.89931*

$$\det\begin{pmatrix} I\_n - \Pi\_1 \mathbf{z} - \dots - \Pi\_p \mathbf{z}^p \end{pmatrix} = \mathbf{0} \tag{6}$$

lie outside the complex unit circle (have modulus greater than one), or, equivalently, if the eigenvalues of the companion matrix have modules less than one. Assuming that the process has been initialized in the infinite past, then a stable VAR (p) process is stationary and ergodic with time invariant means, variances and autocovariances.

If *Yt* is covariance stationary, then the unconditional mean is given by;

$$\mu = \begin{pmatrix} I\_n - \Pi\_1 - \dots \ -\Pi\_p \end{pmatrix}^{-1} \mathbf{c} \tag{7}$$

The mean-adjusted form of the VAR(p) is then;

$$Y\_t - \mu = \Pi\_1(Y\_{t-1} - \mu) + \Pi\_2(Y\_{t-2} - \mu) + \dots + \Pi\_p(Y\_{t-p} - \mu) + \varepsilon\_t \tag{8}$$

The general form of the VAR(p) model with deterministic terms and exogenous variables is given by;

$$Y\_t = \Pi\_1 Y\_{t-1} + \Pi\_2 Y\_{t-2} + \dots + \Pi\_p Y\_{t-p} + \Phi D\_t + G X\_t + \varepsilon\_t \tag{9}$$

where *Dt* represents an *(l* � *1)* matrix of deterministic components, *Xt* represents an *(m* � *1)* matrix of exogenous variables, and ɸ and *G* are parameter matrices.

To be able to estimate the basic VAR(p) model, each equation in the model can be written as;

$$y\_i = Z\pi\_i + e\_i, \ i = 1, \ldots, n \tag{10}$$

where *yi* is a (*T* � *1*) vector of observations on the *i*th equation, *Z* is a *(T* � *k)* matrix with *t*th row given by *Z*<sup>0</sup> *<sup>t</sup>* ¼ 1, *Y*<sup>0</sup> *<sup>t</sup>*�1, … , *<sup>Y</sup>*<sup>0</sup> *t*�*p* , *<sup>k</sup>* <sup>¼</sup> *np* <sup>þ</sup> 1, *<sup>π</sup><sup>i</sup>* is a (k � 1) vector of parameters and *ei* is a *(T* � *1)* error with covariance matrix *<sup>σ</sup>*<sup>2</sup> *<sup>i</sup> IT*. Since the VAR(p) is in the form of a SUR model where each equation has the same explanatory variables, each equation may be estimated separately by ordinary least squares without losing efficiency relative to generalized least squares.
