Update Kyle's Lamba calculation and doc

docs/source/measures/kyle_lambda.rst

@@ -18,7 +18,7 @@ Kyle's Lambda for stock :math:`i` as the slope coefficient
 .. math::
    :label: kylelambda_regression
 
-   r_{i,n} = \delta_{i} + \lambda_{i}\cdot S_{i,n} + \varepsilon_{i,t}
+   r_{i,n} = \lambda_{i}\cdot S_{i,n} + \varepsilon_{i,t}
 
 where for the :math:`n`-th five-minute period, :math:`r_{i,n}` is the
 (percentage) stock return and :math:`S_{i,n}` is the signed square-root dollar

pyproject.toml

@@ -1,6 +1,6 @@
 [project]
 name = "frds"
-version = "2.3.1"
+version = "2.3.2"
 authors = [{ name = "Mingze Gao", email = "[email protected]" }]
 description = "Financial Research Data Services"
 readme = "README.md"

src/frds/measures/_kyle_lambda.py

@@ -13,10 +13,10 @@ def kyle_lambda(returns: np.ndarray, signed_dollar_volume: np.ndarray) -> float:
 
         .. note::
            The return value is the estimated coefficient of :math:`\lambda` in equation
-           :math:numref:`kylelambda_regression` multiplied by 1,000,000. Otherwise, the
-           value would be too small and may cause issues.
+           :math:numref:`kylelambda_regression` multiplied by 1,000,000.
     """
-    returns = np.asarray(returns)
-    signed_dollar_volume = np.asarray(signed_dollar_volume)
-    coeff, _ = np.polyfit(signed_dollar_volume, returns, 1)
-    return coeff * 1_000_000
+    y = np.asarray(returns)
+    x = np.asarray(signed_dollar_volume)
+    x = x[:, np.newaxis]
+    a, _, _, _ = np.linalg.lstsq(x, y, rcond=None)
+    return a[0] * 1_000_000