Financial Insights That Matter
This paper uses the implications of the SVAR model based on monthly data to measure the impact of Bitcoin price shocks on remittances, money multipliers, the USD index, and gold prices; Table 1 provides the descriptive statistics.
Data set
This paper uses monthly datasets. For Bitcoin prices, a series of “Monthly Bitcoin Prices to the US Dollar” is obtained from investing.com, the cryptocurrency section database.
The adoption of Bitcoin is a shift from traditional financial systems, and the marginal spread of this sentiment could lead to a higher demand for gold as a haven, driving up prices. Historical events, such as financial crises in small countries, have occasionally triggered global shifts in gold prices due to perceived risks. As a result, the study adopts “Monthly LBMA Gold Fixing Prices” obtained from investing.com. Further, drawing parallels with other instances where countries adopted unconventional economic policies (e.g., countries with pegged currencies like Taiwan or adopting a currency board), these have often led to reactions in the global commodities and forex markets. For Money Multiplier, a “Monetary and Financial Statistics” series is obtained from the International Monetary Fund database. This study acknowledges that external forces rather than local authorities may influence changes in the monetary multiplier in El Salvador. The country’s reliance on the USD as a legal currency does limit the direct control over monetary policy tools typically available to sovereign nations with their currency. Therefore, this study also uses the “Monthly US Dollar Index (DXY)” obtained from investing.com. The financial market is highly interconnected; even small economies can have outsized effects on global markets through contagion effects. Changes in economic policies, such as adopting Bitcoin, can still influence the perception of the USD in international markets. This can potentially impact investor confidence in the USD, leading to fluctuations in the USD index. Finally, remittance data was collected monthly from the central bank of El Salvador’s macroeconomic data section. Remittances are measured in millions of dollars, focusing on the flow of funds from individuals abroad to recipients in El Salvador. The data explicitly captures the monetary value of remittances in traditional currencies, such as USD.
The monthly sample period between September 10, 2010, and February 30, 2022, has been considered due to consistency in data availability; variables are made stationary for estimation.
Was model
This paper considers a multivariate VAR, as adopted by Momoli (2017), where \({y}_{1{\rm{t}}}\) and depends on different combinations of the previous values of the variables and the error terms
$$\left[\begin{array}{c}{y}_{1{\rm{t}}}\\ {y}_{{\rm{nt}}}\end{array}\right]=\left[\begin{array}{c}{{\rm{\alpha }}}_{1}\\ {{\rm{\alpha }}}_{{\rm{n}}}\end{array}\right]+\left[\begin{array}{cc}{{\rm{\varphi }}}_{11} & {{\rm{\varphi }}}_{12}\\ {{\rm{\varphi }}}_{{ni}} & {{\rm{\varphi }}}_{{ni}}\end{array}\right]\left[\begin{array}{c}{y}_{1{\rm{t}}-1}\\ {y}_{{\rm{nt}}-1}\end{array}\right]+\left[\begin{array}{c}{{\rm{u}}}_{1{\rm{t}}}\\ {{\rm{u}}}_{{\rm{nt}}}\end{array}\right]$$
(1a)
where \({it}\) is a white noise disturbance term with E (\({{\rm{u}}}_{{\rm{it}}}\)) = 0, (i = 1…, n), E (\({{\rm{u}}}_{1{\rm{t}}}{{\rm{u}}}_{{\rm{nt}}}\)) = 0.
We rewrite in matrix form the generic order VAR p as:
$${\rm{y}}_{\rm{t}}=A(L)\,{\rm{y}}_{{\rm{t}}-1}+\varepsilon t$$
(1b)
the stationarity condition is verified if and only if |A(L) − λ| = 0, meaning that λ values are less than 1 in absolute terms. We introduce two dummies (\({b{\mathrm{dummy}}}\) and \({{\mathrm{btdummy}}}\)) to measure the significance of the impact on June 9, 2021, when the government of El Salvador published in the official gazette that made the digital currency Bitcoin legal tender in the country. \({{\mathrm{Btdummy}}}\) measures the impact of the legislation that went into effect on September 7th, 2021. \ ({Bdummy} \) and \({{\mathrm{btdummy}}}\) are significant at 1%.
Analysis of causality according to Granger
In general, interpretative problems arise because (1.1) is a representation in a reduced form that, by its nature, lends little to supporting structural considerations. Granger’s causality analysis aims to evaluate the predictive power of a variable concerning other system variables. According to Granger, if a variable, \({y}_{1}\)assists in improving forecasts of another variable, \({y}_{2}\)then \({y}_{1}\) causes \({y}_{2}\).
Granger causality only means a correlation between the present value of one variable and the past values of others; it does not mean that movements of one variable cause movements of another. There should be caution when interpreting the results of non-causality tests. First, the results are usually sensitive to the set of information used in the application, that is, to the set of strings included in the VAR; there is always the risk of finding “false” causal relationships derived from omitted variables. Causality: By inferring the chronological sequence of movements in the series, one could argue that changes in one variable seem to influence a delay-related variable. This paper rejects the null hypothesis that remittance, the money multiplier, the US Dollar index, and gold prices do not Granger-cause bitcoin price, given that our p value is 0.001.
Impulse response function
We rewrite 1.2 in a compact form as follows, where L is the lag operator:
$${\rm{y}}_{\rm{t}}=A(L){\rm{y}}_{\rm{t}}+{{{{\upepsilon}}}}_{\rm{t}}$$
(1c)
$$A(L)={\rm{A}}_{1}L+{\rm{A}}_{2}{L}^{2}+\ldots +{\rm{A}}_{\rm{P}}{\rm{L}}^{\rm{p}}$$
(1D)
Assuming I-A(L) is invertible, with B(L) = (I-A(L))-lwe can get the moving average representation of the VAR (VMA)
$$\,{\rm{y}}_{\rm{t}}={{\upepsilon}}_{t}+{{\rm{B}}}_{1}{{\upepsilon}}_{t-1}+{\rm{B}}2{{\upepsilon}}_{{\rm{t}}-2}+\ldots +{{\rm{B}}}_{{\rm{s}}}{{\upepsilon}}_{{\rm{t}}-{\rm{s}}}$$
(1e)
We can then interpret as follows matrix \({\beta }_{{\rm{s}}}\):
$${{\rm{B}}}_{{\rm{s}}}=\frac{{\partial }_{{\rm{yt}}+s}}{{\partial }_{{{\upepsilon}}{\rm{t}}}}\,$$
(1f)
In other words, the ij element of \({B}_{{\rm{s}}}\) identifies the consequences of an increase of one unit in the innovations on the jth variable VAR on the value of the ith variable VAR to time t + s, maintaining zero all other innovations in all possible dates between t and t + s. This partial derivation is only meaningful if the impact on the different variables can be assumed to be uncorrelated. Otherwise, if the variables are correlated, there will be an off-diagonal error variance-covariance matrix, resulting in a biased result. The impulse response function describes the effect of a transitory shock (duration is a period) on the VAR variable j on the variable i. However, whether the error terms are correlated is not a valid assumption, as it allows for a current correlation between them.
Structural VAR
The structural VAR with p lags of the n x 1 vector \({y}_{{\rm{t}}}\) of the Eq. (1a) in reduced form is given by:
$$\begin{array}{cc}{\rm{y}}_{\rm{t}}={\sum}_{{\rm{I}}=0}^{{\rm{P}}}{{\rm{A}}}_{{\rm{i}}}{{\rm{y}}}_{{\rm{t}}-1}+{{\rm{Bu}}}_{{\rm{t}}} & ({{\rm{u}}}_{{\rm{t}}}) \sim {\rm{N}}(0,{\rm{I}})\end{array}$$
(1g)
Here, the shocks are perpendicular to each other, interpreted as “raw” shocks with no common cause and unrelated to each other. However, it does not limit the fact that individual shots end up in the same equation. The matrix B cannot be diagonal. This analysis proposes a structural model with traditional functions in the components of different equations that can be related to each other. This correlation happens when multiple equations have at least one impact in common. Since \({it}\) is a response to these innovations, it is worth looking at the impulse response function.
Identification strategy
This paper, similar to Blanchard and Quah (1988), identifies temporary shocks in Bitcoin by imposing long-term constraints to account for the effects of supply and demand shocks. The limit does not affect the long-run impact of the aggregate demand shock on output. The idea behind these constraints is the existence of a vertical supply curve. The identification restriction is given by \({{\rm{\varphi }}}_{12}\left(1\right)=0\) in Eq. 1a. The VAR can be implemented as follows:
$$\,\left[\begin{array}{c}{\Delta y}_{{\rm{t}}}\\ {y}_{{\rm{nt}}}\end{array}\right]=\left[\begin{array}{cc}{{\rm{A}}}_{11}(L) & {{\rm{A}}}_{12}(L)\\ {{\rm{A}}}_{{ni}}(L) & {{\rm{A}}}_{{ni}}(L)\end{array}\right]\left[\begin{array}{c}{{\rm{u}}}_{1{\rm{t}}}\\ {{\rm{u}}}_{{\rm{nt}}}\end{array}\right]$$
(1h)
Where E (\({{\rm{u}}}_{{\rm{t}}}{{\rm{u}}{\prime} }_{{\rm{t}}}\)) = \(\Omega\). Let S = chol (\({\rm{A}}(1)\Omega {\rm{A}}(1){\prime} )\) and K = \ ({\ rm {a}} (1) \)-1S. The identified shocks are \({w}_{{\rm{t}}}=K\)-1\({{\rm{u}}}_{{\rm{t}}}\)and the resulting impulse response to structural shocks is \ (\ Verphi ({\ rm {l}}) \) = \(A\)(L)K. Notice that the restrictions are satisfied in the equation below:
$$ \ varphi (1) = {\ rM {a}} (1) {\ rm {k}} $$
$$ = {\ rm {a}} (1) {\ rm {a}} (1) {{-1} {\ rm {s}} $$
which is lower and triangular, implying that \ ({\ rM {\ varphi}}} _ {12} \)(1) = 0 in Eq. 1a.
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