### Methodology

This research article estimates herding behaviour by describing cross-sectional standard deviation (CSSD) model by Christie and Huang (1995) and cross-sectional absolute deviation (CSAD) model by Chang et al. (2000). Chang et al. (2000) proposed absolute values of returns for removing the effect of outliers among observations in the model of Christie and Huang (1995). After this, Hwang and Salmon (2004) suggested a state-space model to estimate herding on macro-levels. But their approach requires low-frequency data. In this research, we prefer to use daily observations of currencies, i.e. high-frequency data. Besides, Demirer et al. (2010) considered model of Chang et al.(2000) as effective as Hwang and Salmon (2004)’s model. Therefore, this research paper uses CSAD method. Herding behaviour is estimated by using Newey and West’s heteroscedasticity and autocorrelation consistent standard errors to solve any multicollinearity problem (Newey and West 1987).

### Data collection

Christie and Huang (1995) mentioned that herding behaviour is displayed in high-frequency data. Therefore, this study utilizes daily prices of top six traded currencies in Pakistan. Selected currencies are on the top of the list in all major exchange platforms. Prices are taken from January 1, 2015 to December 31, 2019, generating a total of 1305 observations. Data of Saturday and Sunday were excluded as the trading is closed on weekend. Currencies under study are: (1) Pound Sterling (UK), (2) Dollar (US), (3) Euro (European Union), (4) Yen (Japan), (5) AED (UAE), (6) Riyal (Saudi Arabia).

### CSSD and CSAD herding models

In 1995, Christie and Huang proposed the following cross-sectional standard deviation (CSSD) method to capture the dispersion among assets:

$$CSSD_{t } = sqrt {frac{{{Sigma }_{i = 1 }^{N} left( {R_{i,t} – R_{m,t} } right)^{2} }}{N – 1}} ,$$

(1)

where ({R}_{i,t}) represents stock return of the firm *i* at time *t*, ({R}_{m,t}) is the cross-sectional average return of the *N* returns in the equally weighted market portfolio at time *t*, and *N* is the number of stocks in the market portfolio.

CSSD method becomes sensitive to outliers. To solve this issue, Chang et al. (2000) proposed cross-sectional absolute deviation (CSAD) instead of CSSD. CSAD model uses absolute values of returns to remove the effects of outliers in the observations. This method is represented as:

$$CSAD_{t } = frac{1}{N} mathop sum limits_{i = 1}^{N} left| {R_{i,t} – R_{m,t} } right|.$$

(2)

In this approach, absolute values of returns are calculated. ({R}_{m,t}) represents the average of returns in market portfolio which are generated by investors’ behaviour. Difference of ({R}_{i,t}) and ({R}_{m,t}) specifies the dispersion of returns. Therefore, CSAD equation capturing the dispersion and herding effects as suggested by Chang et al. (2000) is given below:

$$CSAD_{(t)} = alpha + gamma_{1} |R_{m,t} | + gamma_{2} (R_{(m,t)}^{2} ) + varepsilon_{t} ,$$

(3)

where (left|{R}_{m,t}right|) is the absolute equally weighted market return, and ({R}_{m,t}^{2}) is the squared market return.

This equation identifies herding behaviour by establishing a link between CSAD and ({R}_{m,t}). Presence of herding behaviour among stocks or currencies is manifested by statically significant negative coefficient ({gamma }_{2}). It means high correlation between individual returns of currencies. If there is no presence of herding behaviour among returns, there will be an increase in dispersion.

### Asymmetric effects in herding models

Previous studies of Christie and Huang (1995), Chang et al. (2000), Demirer et al. (2010), Rompotis (2018), and Ballis and Drakos (2019) discussed in their studies that cross-sectional average of returns varies in high and low-volume trading days. Therefore, this study also examines effect of asymmetry in returns of currencies in market.

Asymmetry between the relationship of CSAD and market returns can be estimated by the following two equations, as proposed by Chang et al. (2000):

$$CSAD_{t}^{{UP}} = alpha + gamma _{1}^{{UP}} |R_{{m,t}}^{{UP}} | + gamma _{2}^{{UP}} (R_{{m,t}}^{{UP}} )^{2} + varepsilon _{t} {mkern 1mu} ,if{mkern 1mu} R_{{m,t}} {text{ > }}0,$$

(4)

$$CSAD_{t}^{{DOWN}} = alpha + gamma _{1}^{{DOWN}} |R_{{m,t}}^{{DOWN}} | + gamma _{2}^{{DOWN}} (R_{{m,t}}^{{DOWN}} )^{2} + varepsilon _{t} ,if{mkern 1mu} R_{{m,t}} {text{ < 0}},$$

(5)

where (left|{R}_{m,t}^{UP}right|) ((left|{R}_{m,t}^{DOWN}right|)) is the equally weighted average return of the *N* currencies in the market available on day *t* when the return is positive (negative). Similarly, ({CSAD}_{t}^{UP}) is the ({CSAD}_{t}) on the day *t*, where ({R}_{m,t}) is positive and ({CSAD}_{t}^{DOWN}) is the ({CSAD}_{t}) for the day *t*, where ({R}_{m,t}) is negative.

For the purpose of this paper, firstly daily return of each currency is calculated as suggested by Ballis and Drakos (2019):

$$R_{i,t} = ln frac{{P_{i,t} }}{{P_{i,t – 1} }} ,$$

(6)

where *i* represents each currency, *t* is the time period and *P* shows the closing price of each selected currency.

CSAD is calculated using Eq. (2), and herding behaviour is analysed by regression Eq. 3. Similarly, Eqs. (4) and (5) are also estimated when market returns are high and low. Analysis is run in STATA using Newey–West standard errors and other time-series techniques.

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