# Monotonicity in Asset Returns

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Many theories in finance imply monotonic patterns in expected returns and other financial variables.

- the liquidity preference hypothesis predicts higher expected returns for bonds with longer times to maturity
- the Capital Asset Pricing Model (CAPM) implies higher expected returns for stocks with higher betas.
- standard asset pricing models imply that the pricing kernel is declining in market returns

The paper proposes several ways to test for monotonicity in financial variables and compares the proposed tests with extant alternatives such as $t$-tests, Bonferroni bounds, and multivariate inequality tests.

Denote the expected returns by $\mu = (\mu_0,\ldots,\mu_N)’$, and define the associated return differentials as $\Delta_i=\mu_i-\mu_{i-1}$.

Test

\[H_0: \Delta \le 0\qquad \text{vs}\qquad H_1:\Delta > 0\]where $H_1$ can be rewritten as

\[H_1:\min_{i=1,\ldots,N}\Delta_i > 0\,,\]which is called monotonic relation (MR) test. Because the test focuses on the smallest deviation from the null hypothesis, the power would grow relatively slowly as the sample size expands.

Propose an Up and a Down statistic that account for both the frequency, magnitude and direction of deviations from a flat pattern.

\[J_T^- = \sum_{i=1}^N\vert \hat\Delta_i\vert 1(\hat\Delta_i < 0)\\ J_T^+ = \sum_{i=1}^N\vert \hat\Delta_i\vert 1(\hat\Delta_i > 0)\]Wolak (1989) tests (weak) monotonicity under the null and specifies the alternative as non-monotonic.

- one potential drawback is that limited power makes it difficult to reject the null hypothesis

A naive approach is to conduct a set of pairwise $t$-tests to see if $\Delta_i$ is positive for each $i=1,\ldots,N$. Use Bonferroni bound to combine $N$ tests.