Story roadmap

Question → Statistical Idea → Mathematics → R → Visualization → Insight

• Data: daily closing prices of the DAX, SMI, CAC and FTSE indices, 1860 trading days in the 1990s (R dataset EuStockMarkets)
• Variables: daily log returns of CAC (response) and DAX (predictor)
• Tool: simple linear regression, where the slope is the market beta

Investors care about percentage changes, not price levels, and prices are non-stationary. So we convert prices to daily log returns.

• Mean daily CAC return: 0.044%
• Correlation between DAX and CAC returns: r = 0.734

In finance, the beta of an asset measures how strongly it moves with a benchmark.

• \(\beta = 1\): moves one-for-one with the benchmark.
• \(\beta > 1\): amplifies the benchmark’s moves (more aggressive).
• \(\beta < 1\): dampens the benchmark’s moves (more defensive).
• The intercept (often called alpha) is the average return not explained by the benchmark.
• The residual is the part of the return that is specific to the asset.

Here we treat DAX as the benchmark and ask for the CAC’s beta.

The simple linear regression (market) model:

\[R_{\text{CAC},t} = \alpha + \beta\, R_{\text{DAX},t} + \varepsilon_t, \qquad \varepsilon_t \stackrel{iid}{\sim} N(0,\sigma^2)\]

Least squares minimizes

\[SSE = \sum_{t=1}^{n}\left(r_{\text{CAC},t} - \alpha - \beta\, r_{\text{DAX},t}\right)^2\]

The estimated beta is also a ratio of covariance to variance:

\[\hat\beta = \frac{\widehat{\mathrm{Cov}}(R_{\text{DAX}},R_{\text{CAC}})}{\widehat{\mathrm{Var}}(R_{\text{DAX}})} = r\,\frac{s_{\text{CAC}}}{s_{\text{DAX}}}\]

A \(95\%\) confidence interval for \(\beta\):

\[\hat\beta \pm t_{n-2,\,0.975}\; SE(\hat\beta), \qquad SE(\hat\beta)=\frac{\hat\sigma}{\sqrt{\sum_t (x_t-\bar x)^2}}\]

To test whether the CAC moves with the DAX at all, we test \(H_0:\beta=0\) using

\[t=\frac{\hat\beta}{SE(\hat\beta)} \sim t_{n-2} \quad \text{under } H_0\]

To ask whether the CAC is more or less volatile than the DAX, test \(H_0:\beta=1\) with \(t=(\hat\beta-1)/SE(\hat\beta)\).

library(ggplot2)
library(plotly)

# Prices -> daily log returns
prices  <- as.data.frame(EuStockMarkets)
returns <- as.data.frame(apply(log(prices), 2, diff))

# Beta of CAC relative to DAX
fit <- lm(CAC ~ DAX, data = returns)
summary(fit)
confint(fit)

# Test H0: beta = 1
b  <- coef(summary(fit))["DAX", ]
tt <- (b["Estimate"] - 1) / b["Std. Error"]
2 * pt(-abs(tt), df = fit$df.residual)

ggplot(returns, aes(DAX, CAC)) +
  geom_point(alpha = 0.3, color = "#8C1D40") +
  geom_smooth(method = "lm", color = "#FFC627")

Each dot is one trading day. The upward slope of the fitted line is the beta.

• Estimated beta: 0.786, 95% CI (0.753, 0.82).
• A 1% rise in the DAX goes with about a 0.79% rise in the CAC, on average.
• \(R^2 = 0.539\): the DAX explains about 54% of the CAC’s daily variation.
• Test of \(H_0:\beta=1\): p-value = 2.7^{-35}.

Residuals are centered at zero, but notice that their spread changes over time. Calm and volatile periods cluster, which is a well-known feature of financial returns (volatility clustering) and a limit of the constant-variance assumption.

Rotate and zoom! The points form a tilted cloud: when DAX and SMI move up, so does CAC. Color shows the FTSE tends to move along as well.

• Question: How does the CAC respond to moves in the DAX?
• Answer: The two markets move closely together, with an estimated beta of about 0.79, and the relationship is highly statistically significant.
• Strength: The DAX alone explains about 54% of the CAC’s daily variation.
• Caution: Residual variance changes over time (volatility clustering), so standard errors from this simple model may be too optimistic.
• Next step: Add other markets (multiple regression) or model changing volatility (GARCH).

• R dataset EuStockMarkets: daily closing prices of major European stock indices, 1991-1998 (Erste Bank AG, Vienna), distributed with base R.
• Packages: ggplot2, plotly (R).